Squarefree powers of closed neighborhood ideals

SQUAREFREE PO WERS OF CLOSED NEIGHBORHOOD IDEALS MARIE AMALORE NAMBI AND A YESHA ASLOOB QURESHI Abstract. In this article, we c haracterize all trees whose highest non-v anishing squarefree p o wer of the closed neighborho o d ideal is component wise linear. In addition, we in vestigate the Castelnuo v o-Mumford regularit y of the ν -th squarefree p ow er of the closed neigh b orho od ideal of trees and sho w that this num ber can b e arbitrarily larger than the degree of the ideal. Finally , w e giv e a form ula for the regularit y of ν -th squarefree p ow er of the closed neighborho o d ideal of caterpillar graphs. 1. Introduction The core of com binatorial comm utative algebra lies in the study of the interpla y b et w een the algebraic prop erties of an ideal and the com binatorial structure of the underlying ob ject. In particular, squarefree monomial ideals are naturally connected with combinatorial ob jects such as simplicial complexes, graphs, and clutters, oﬀering the study of their algebraic and homological prop erties via the asso ciated com binatorial data. The existence of a linear resolution has b een one of the most extensively studied homological inv arian ts. Recall that a line ar r esolution is a minimal free resolution in which eac h diﬀerential map is represen ted by matrix that ha v e entries in the set of linear forms. These ideals are w ell-b eha ved and computationally simpler to work with than most others. A remark able result in this direction is due to F r¨ ob erg [4] who establishes that the edge ideal of a graph admits a linear resolution if and only if the graph is co-c hordal. Herzog and Hibi [7] in tro duced the notion of comp onen twise linear ideals, whic h form the next b est class after ideals with linear resolutions. Let I ⊆ S = K [ x 1 , . . . , x n ] b e a graded ideal. Recall that I is said to b e c omp onentwise line ar if, for each degree k , the ideal generated by all homogeneous elemen ts of I of degree k has a linear resolution. It is kno wn from [7] that a Stanley- Reisner ideal I is sequen tially Cohen-Macaula y if and only if its Alexander dual is component wise linear. F or further details on component wise linearit y , w e refer the reader to the surv ey article [6]. In this context, we study the component wise linearit y of the highest non-v anishing squarefree p o wer of the closed neighborho o d ideal of trees is of particular imp ortance. Sharifan and Moradi [17] introduced the close d neighb orho o d ide al N I ( G ) of a ﬁnite simple graph G , a class of squarefree monomial ideals generated by monomials corresponding to the closed neigh b orho o ds of the v ertices of G . This ideal can b e view ed as the facet ideal of the neighbor- ho o d complex N ( G ) of G . F rom the p erspective of the Stanley–Reisner complex, its Alexander dual is identiﬁed with the dominance complex. In recen t years, sev eral algebraic prop erties of 2020 Mathematics Subje ct Classiﬁc ation. 13D02, 05E40, 05E45, 05C70. Key wor ds and phr ases. Squarefree p ow er, closed neighborho o d ideal, matc hing, comp onent wise linear, Casteln uov o-Mumford regularity . The authors are supp orted by the Scien tiﬁc and T echnological Researc h Council of T urkey T ¨ UB ˙ IT AK under the Gran t No: 124F113, and thankful to T ¨ UB ˙ IT AK for their supp orts. 1 2 closed neigh b orho o d ideals such as regularit y , pro jectiv e dimension, Cohen-Macaulayness, min- imal irreducible decomp ositions, normally torsion-free prop ert y , and p ersistence property , hav e b een studied in [3, 12, 13, 14, 15, 16, 17, 18]. T o the b est of our kno wledge, this is the ﬁrst study that fo cuses on squarefree p ow er of these ideals in the literature. Let I b e a monomial ideal. The k -th squarefree p ow er of I , denoted by I [ k ] , is the ideal generated by all squarefree monomials b elonging to I k . F rom a combinatorial p ersp ective, the squarefree p o w ers of I are closely related to the matching theory of simplicial complexes or clutters. Sp eciﬁcally , the minimal generators of I [ k ] are k -matchings of the associated simplicial complex. The squarefree p ow ers of an ideal carry imp ortan t information ab out its ordinary p o wers. In particular, if I [ k ] do es not hav e a comp onent wise linear, then I k do es not hav e a comp onen twise linear (see [8, Lemma 4.4]). Bigdeli, Herzog, and Zaare-Nahandi [2, Theorem 5.1] sho wed that the highest non-v anishing squarefree p ow er of the edge ideal of a graph has linear quotients. Kam b eri, Na v arra, and the second author of this article [13] show ed that the highest non-v anishing squarefree pow er of the facet ideal of a simplicial tree with the intersection prop ert y has linear quotien ts. How ever, they further demonstrated that this is not true in general for simplicial trees. A natural question that arises in this context is the following. Question 1.1. Is the highest non-v anishing squarefree p o wer of the closed neighborho o d ideal of a tree is comp onen twise linear? Ho w ever, Question 1.1 do es not hold in general as observ ed in Example 2.6. Motiv ated by this, w e provide a combinat orial c haracterization of trees whose highest non-v anishing squarefree p o wer of the closed neigh borho o d ideal is comp onen t wise linear (see Theorems 4.1, 4.2, and 4.13). The second part of this article is dedicated to study the Castelnuo v o-Mumford regularity (regularit y) of the highest non-v anishing squarefree p ow er of the closed neigh b orho o d ideal of graphs. Sharifan and Moradi in [17] inv estigated the regularit y of the closed neigh b orho o d ideal of v arious classes of graphs. In particular, they obtained the exact form ula for the regularit y of the closed neighborho o d ideal of path graphs, generalized star graphs, complete r -partite graphs and m -bo ok graphs. Moreov er, for a forest, they obtained a low er b ound for the regularity of the closed neigh b orho o d ideal in terms of the matching n um b er. Chakrab orty et al. [3] obtained a precise form ula for the regularity of the closed neighborho o d ideal of forests in terms of the matc hing n um b er. F urthermore, they sho w ed that, for any graph G , the matc hing n um b er of G pro vides a low er b ound for reg ( S/ N I ( G )). A natural question is whether the diﬀerence b et w een the regularit y of the highest non-v anishing squarefree p ow er of the closed neighborho o d ideal of a graph and the degree of the ideal can become arbitrarily large. More precisely , w e may ask the follo wing question. Question 1.2. Given an in teger m . Is there exists a connected graph such that reg( S/ ( N I ( G ) [ ν ] )) − deg( N I ( G ) [ ν ] ) = m ? W e give a p ositiv e answer to Question 1.2 (see Theorems 5.2 and 5.4). This indicates that de- termining the regularit y of the highest non-v anishing squarefree pow er of the closed neigh borho o d ideal for an arbitrary tree is a hard problem. Therefore, w e conclude by fo cusing on a speciﬁc class of trees, namely caterpillar graphs (see Theorem 5.5). 3 2. Required Ingredients W e ﬁrst recall some basic deﬁnitions, notations, and results from graph theory and comm utativ e algebra that will b e used throughout the subsequent sections. Graph Theory. Let G b e a ﬁnite simple graph with v ertex set V ( G ) and edge set E ( G ). F or a subset U ⊆ V ( G ), the induc e d sub gr aph of G on U , denoted b y G [ U ], is the graph with v ertex set U and edge set E ( G [ U ]) = { e ∈ E ( G ) : e ⊆ U } . A sequence of vertices v 1 , . . . , v k in G is said to b e a p ath of length k − 1 if { v i , v i +1 } ∈ E ( G ) for all 1 ≤ i ≤ k − 1. Given an y u, v ∈ V ( G ), we set the length of the shortest path connecting u and v by d ( u, v ). Moreov er, a sequence of v ertices v 1 , . . . , v k in G is said to b e a cycle of length k if { v i , v i +1 } ∈ E ( G ) for all 1 ≤ i ≤ k where v 1 = v k +1 . A graph G is called a for est if it do es not contain a cycle as an induced subgraph. A tr e e is a connected forest. F or a v ertex u ∈ V ( G ), let G \ u denote the induced subgraph of G on the v ertex set V ( G ) \ { u } . F or a vertex u ∈ V ( G ), the set of neigh b ors of u , denoted by N G ( u ), is the set { v ∈ V ( G ) : { u, v } ∈ E ( G ) } . The degree of the v ertex u in G is the cardinality of N G ( u ), and is denoted by deg G ( u ). The set of closed neigh b ors of u , denoted by N G [ u ], is the set N G ( u ) ∪ { u } . Neigh b orho o d Complex. A simplicial complex ∆ on the v ertex set V (∆) = [ n ] is a non-empt y collection of subsets of V (∆) such that if F ∈ ∆ and G ⊆ F , then G ∈ ∆. The elements of ∆ are called faces. F or any F ∈ ∆, the dimension of F is one less than the cardinalit y of F . The maximal faces of ∆ with resp ect to inclusion are called facets, and the set of all facets of ∆ is denoted by F (∆). A sub collection of a simplicial complex ∆ is a simplicial complex whose facet set is a subset of the facet set of ∆. A facet F of a simplicial complex ∆ is called a leaf if either F is the only facet of ∆ or there exists a facet G ∈ ∆ suc h that G  = F and F ∩ G ⊇ F ∩ H for all facets H  = F of ∆. A simplicial complex ∆ is a simplicial for est if every nonempty sub collection of ∆ has a leaf. A connected simplicial forest is called a simplicial tr e e . The closed neighborho o d complex of a graph G , denoted by N ( G ), is the simplicial complex on vertex set V ( G ) deﬁned by N ( G ) = { F ⊆ V ( G ) : there exists vertex u such that F ⊆ N G [ u ] and N G [ v ] ⊆ F for all v ∈ V ( G ) \ u } . Set [ n ] = { 1 , . . . , n } . Belo w, w e present an example of the closed neighborho od complex of a graph. Example 2.1. Let G b e a graph on [10] with edge set E ( G ) = {{ 1 , 2 } , { 2 , 3 } , { 3 , 4 } , { 4 , 5 } , { 5 , 6 } , { 6 , 7 } , { 7 , 8 } , { 8 , 9 } , { 5 , 10 }} . Then the facets of the closed neigh b orho o d complex N ( G ) are F ( N ( G )) = {{ 1 , 2 } , { 5 , 10 } , { 8 , 9 } , { 2 , 3 , 4 } , { 3 , 4 , 5 } , { 5 , 6 , 7 } , { 6 , 7 , 8 }} . As shown in [1], the following remark recalls that the closed neighborho o d complex of a tree is a simplicial forest. Remark 2.2. [1, Chapter 5, Example 3, page 174] [9, Theorem 3.2] Let G b e a tree. Then the closed neighborho o d complex N ( G ) is totally balanced. Equiv alen tly , N ( G ) is a simplicial forest. 4 Squarefree p o w ers and matc hings. Let S = K [ x 1 , . . . , x n ] be a polynomial ring in n v ariables o v er a ﬁeld K . Let I b e a monomial ideal of S . Let G ( I ) denote the minimal monomial generating set of a monomial ideal I . W e denote by I [ a ] the ideal generated by the squarefree monomials b elonging to I a , by I ⟨ a ⟩ the ideal generated b y all homogeneous elemen ts of degree a b elonging to I , and b y deg ( I ) the largest degree among the minimal generators of I . F or a subset A ⊆ [ n ] we asso ciate a squarefree monomial x A := Q u ∈ A x u ∈ S , and the set A is called the supp ort of x A , denoted by supp( x A ). Let ∆ be a simplicial complex. The facet ideal of ∆ is deﬁned as I (∆) = ( x F : F ∈ F (∆)) ⊂ S . The facet ideal of the closed neigh b orho o d complex of a graph G is called the closed neighborho o d ideal of G , denoted b y N I ( G ). Example 2.3. Let G b e the graph as in Example 2.1, and let N ( G ) b e its closed neighborho o d complex. Then the closed neighborho od ideal of G is N I ( G ) = ( x 1 x 2 , x 5 x 10 , x 8 x 9 , x 2 x 3 x 4 , x 3 x 4 x 5 , x 5 x 6 x 7 , x 6 x 7 x 8 ) ⊂ S = K [ x 1 , . . . , x 10 ] . A matching of ∆ is a set of pairwise disjoint facets of ∆. F or a matching M of ∆, w e set V M = { u ∈ V (∆) : u ∈ F and F ∈ M } . A matching consisting of k facets is referred to as a k -matc hing. A k -matc hing is called maximal, if ∆ do es not admit any ( k + 1)-matching. The matching numb er of ∆ is the size of a maximal matc hing of ∆ and is denoted by ν (∆). Let G b e a graph and M b e a k -matc hing of N ( G ). F or a subgraph H of G , w e denote b y M H = { F ∈ M : F ⊆ V ( H ) } , the sub collection of facets of M whose vertex sets are con tained in V ( H ). W e demonstrate the ab ov e deﬁnitions with the following example. Example 2.4. Let G b e the graph describ ed in Example 2.1, and let N ( G ) denote its closed neigh b orho o d complex. Then M = {{ 1 , 2 } , { 5 , 10 } , { 8 , 9 }} is a maximal matc hing of N ( G ), and hence ν ( N ( G )) = 3. Let H = G \ { 5 } b e the induced subgraph of G . Then M H = {{ 1 , 2 } , { 8 , 9 }} . 2.1. Comp onent wise Linearit y and Linear Relations. Let S = K [ x 1 , . . . , x n ] b e a p olyno- mial ring ov er a ﬁeld K and I b e a homogeneous ideal of S . Let F • b e a minimal graded S -free resolution of I : F • : ⊕ j ∈ Z S ( − j ) β p,j ( I ) ϕ p − → ⊕ j ∈ Z S ( − j ) β p − 1 ,j ( I ) − → · · · − → ⊕ j ∈ Z S ( − j ) β 0 ,j ( I ) ϕ 0 − → I − → 0 , where p ≤ n . The num bers β i,j ( I ) are called the ( i, j )-th graded Betti num b ers of I . The Castelnuovo-Mumfor d r e gularity or simply r e gularity of I ov er S , denoted by reg ( I ), is deﬁned as reg( I ) : = max { j : β i,i + j ( I )  = 0 , for some i } . Moreo v er, we hav e β i,j ( I ) = β i +1 ,j ( S/I ) for all i ≥ 0 and j ∈ Z . The following lemma is a key ingredient that will b e used rep eatedly in Section 5. Lemma 2.5. [11, Lemma 3.1] L et I b e a homo gene ous ide al of S and f b e a de gr e e d monomial of S . Consider 0 → S/ ( I : f )( − d ) → S / ( I ) → S/ ( I + f ) → 0 the short exact se quenc e. Then one has reg( S/ ( I + f )) ≤ max { reg ( S/ ( I : f )) + d − 1 , reg( S/I ) } . 5 The e quality holds if reg ( S/ ( I : f )) + d  = reg ( S/I ) . A homogeneous ideal I ⊂ S has a d - line ar r esolution if β i,j ( I ) = 0 for all i ≥ 0 and for j  = i + d . A homogeneous ideal I ⊂ S has line ar quotients if there is a system of minimal homogeneous generators u 1 , u 2 , . . . , u n suc h that the colon ideal ( u 1 , . . . , u i − 1 ) : u i is generated by linear forms for eac h i = 2 , . . . , n . F rom [10, Prop osition 8.2.1], it follows that if I has linear quotients and generated in degree d then I has a d -linear resolution. A homogeneous ideal I ⊂ S is said to b e c omp onentwise line ar if I ⟨ j ⟩ has a linear resolution for all j . F rom [10, Theorem 8.2.15], it follo ws that if I has linear quotien ts then I is comp onent wise linear. An ideal I ⊂ S , generated b y homogeneous elements of degree d , is said to b e line arly r elate d if β 1 ,j ( I ) = 0 for all j  = 1 + d . In particular, if I is not linearly related then I is not comp onen t wise linear. It is known from [2, Theorem 5.1] that the ν -th squarefree p o w er of the edge ideal of a graph has linear quotients, and hence is comp onent wise linear. How ev er, this result do es not hold in general for monomial ideals. In the following example, we show that the ν -th squarefree p o w er of the closed neighborho o d ideal of a graph do es not need to b e comp onen t wise linear. Example 2.6. Let G b e the graph giv en in Example 2.1, and let N ( G ) denote its closed neigh- b orho o d complex. Then ν ( N ( G )) = 3. Macaulay2 [5] computation sho ws that ( N I ( G ) [3] ) ⟨ 7 ⟩ do es not hav e a linear resolution, therefore the ideal N I ( G ) [3] is not comp onen twise linear. In the following, we recall a useful to ol from [2] to inv estigate the linearly related prop ert y of monomial ideals. Let I b e a monomial ideal generated in degree d . In [2], the authors asso ciated a graph G I to I as follows: V ( G I ) = G ( I ), and { u, v } ∈ E ( G I ) if and only if deg(lcm( u, v )) = d + 1. Moreo v er, for all u, v ∈ G ( I ), the induced subgraph of G I on the vertex set { w ∈ V ( G I ) : w divides lcm( u, v ) } is denoted b y G ( u,v ) I . Theorem 2.7. [2, Corollary 2.2] L et I b e a monomial ide al gener ate d in de gr e e d. Then I is line arly r elate d if and only if for al l u, v ∈ G ( I ) ther e is a p ath in G ( u,v ) I c onne cting u and v . 3. Some lemmas on ma tching of simplicial trees In this section, w e establish sev eral combinatorial prop erties of k -matchings of simplicial trees, whic h will b e used in the subsequent section. Construction 3.1. Let ∆ be a simplicial complex, and let M and N be tw o distinct k -matchings of ∆. Then there exist ˆ M ⊂ M and ˆ N ⊂ N such that ˆ M ∩ ˆ N = ∅ and M \ ˆ M = N \ ˆ N . That is, ˆ M and ˆ N are obtained b y removing the common elemen ts of M and N . Then ˆ M and ˆ N form a ˆ k -matching of ∆ with ˆ k ≤ k . Let ˆ M = { F 1 , . . . , F ˆ k } and ˆ N = { G 1 , . . . , G ˆ k } . W e construct the bipartite graph B ( M ,N ) on the v ertex set ˆ M ⊔ ˆ N and the edge set {{ F i , G j } : F i ∩ G j  = ∅} . The following remark follows from the construction ab o v e. Remark 3.2. F ollowing the notation in Construction 3.1, if F i ⊂ V ˆ N = S ˆ k j =1 G j , then the degree of F i in B ( M ,N ) is at least 2. W e next show that the graph B ( M ,N ) asso ciated to t w o k -matc hings of a simplicial tree has a particularly simple structure. 6 Lemma 3.3. L et ∆ b e a simplicial tr e e, and let M and N b e two distinct k -matchings of ∆ . Then the bip artite gr aph B ( M ,N ) deﬁne d in Construction 3.1 is acyclic. Pr o of. Supp ose, to the con trary , that B ( M ,N ) has a cycle C = F 1 , G 1 , F 2 , . . . , F q , G q , F 1 of minimal length q ≥ 4, after reordering the vertices if necessary . Consider the sub complex ∆ ′ = ⟨ F 1 , . . . , F q , G 1 , . . . , G q ⟩ ⊆ ∆. Then ∆ ′ is also a simplicial forest with a leaf, sa y G 1 . It follo ws from the construction of B ( M ,N ) that G 1 ∩ F 1 and G 1 ∩ F 2 are non-empt y , and b y the minimalit y of q , G 1 ∩ F i = ∅ for all 3 ≤ i ≤ q . Since M is a matching, F 1 and F 2 are disjoin t, and therefore G 1 ∩ F 1 and G 1 ∩ F 2 are also disjoin t. This implies that G 1 is not a leaf of ∆ ′ , a con tradiction to ∆ b eing a simplicial tree. □ Prop osition 3.4. L et ∆ b e a simplicial tr e e, and let M and N b e two k -matchings of ∆ . Supp ose that V M ⊆ V N . Then M = N . Pr o of. F ollo wing Construction 3.1, consider the bipartite graph B ( M ,N ) . Since V M ⊆ V N , by the construction of ˆ M and ˆ N it follo ws that V ˆ M ⊆ V ˆ N . In particular, for eac h F i ∈ ˆ M , w e ha v e F i ⊆ V ˆ N . Then, following Remark 3.2, the degree of F i is at least 2 for all 1 ≤ i ≤ ˆ k . Consequen tly , the num ber of edges of B ( M ,N ) is at least 2 ˆ k . On the other hand, since B ( M ,N ) is acyclic by Lemma 3.3, an acyclic graph on 2 ˆ k vertices has at most 2 ˆ k − 1 edges. This is a con tradiction. Hence M = N , as desired. □ As an immediate consequence, we obtain the follo wing description of the elemen ts of G ( N I ( T ) [ ν ] ), where T is a tree. Corollary 3.5. L et T b e a tr e e, and let N ( T ) b e the close d neighb orho o d c omplex with F ( N ( T )) = { F 1 , . . . , F n } . Then e ach u ∈ G ( N I ( T ) [ k ] ) c an b e written uniquely as u = x F i 1 · · · x F i k , wher e M = { F i 1 , . . . , F i k } is a k -matching of N ( T ) . W e recall a useful result from [13] related to the matchings of a simplicial tree. Theorem 3.6. [13, Lemma 3.5] L et ∆ b e a simplicial tr e e. F urther, let M = { F 1 , . . . , F k } and N = { G 1 , . . . , G k } b e two k -matchings of ∆ . Then ther e exist i, j ∈ { 1 , . . . , k } such that F i ∩ G s = ∅ for al l s  = j . Remark 3.7. Moreov er, by remo ving the common facets from M and N in Theorem 3.6, one ma y conclude that there exist i, j ∈ { 1 , . . . , k } such that F i  = G j and F i ∩ G s = ∅ for all s  = j . Moreo v er, if k is the matching num b er of ∆, then F i ∩ G j  = ∅ . The follo wing lemmas describ e combinatorial prope rties of tw o ν -matc hings of N ( T ), and they pla y a k ey role in the pro of of the Theorem 4.13. In particular, w e compare the n um b ers of facets con tained in the comp onents of T \ { u } . Lemma 3.8. L et T b e a tr e e and let u b e a vertex of T . L et T 1 , . . . , T d b e the c onne cte d c omp onents of the gr aph T \ { u } . Supp ose that M and N ar e two ν -matchings of N ( T ) . Then, for e ach i = 1 , . . . , d , one has | M T i | − 1 ≤ | N T i | ≤ | M T i | + 1 . 7 Pr o of. By symmetry , it suﬃces to pro v e the inequalities for i = 1. First, supp ose that a = | N T 1 | < | M T 1 | − 1 . The set M T 1 ∪ S d i =2 N T i forms a matc hing of N ( T ), since all its facets are con tained in pairwise disjoint comp onen ts of T \ { u } . Moreov er, N T [ u ] is the unique facet of N ( T ) that in tersects more than one comp onent of T \ { u } . Hence, among the facets of N , at most one in tersects more than one comp onent, and therefore P d i =2 | N T i | ≥ ν − ( a + 1) . It follows that   M T 1 ∪ d [ i =2 N T i   = | M T 1 | + ν − a − 1 > ν , whic h contradicts the fact that ν is the matching num ber of N ( T ). Therefore, | N T 1 | ≥ | M T 1 | − 1 . Similarly , by interc hanging the roles of M and N , w e obtain | M T 1 | ≥ | N T 1 | − 1 which gives | M T 1 | + 1 ≥ | N T 1 | . □ Lemma 3.9. Under the assumptions of L emma 3.8, additional ly supp ose that N T [ u ] ∈ M . Then the fol lowing hold. (i) If N T [ u ] do es not interse ct any fac et of N T i for some i , then | M T i | = | N T i | . (ii) If N T [ u ] / ∈ N and u ∈ V N , then either | M T i | = | N T i | for al l i , or ther e exist distinct indic es i, j such that | N T i | = | M T i | + 1 , | N T j | = | M T j | − 1 , and | N T k | = | M T k | for al l k  = i, j . (iii) If u / ∈ V N , then ther e exists an index i such that N T i c ontains a fac et that c ontains a neighb or of u , and | N T i | = | M T i | + 1 , while | N T j | = | M T j | for al l j  = i . Pr o of. Since N T [ u ] ∈ M and every other facet of M b elongs to exactly one M T i , we ha ve M = { N T [ u ] } ⊔ M T 1 ⊔ · · · ⊔ M T d , ν = 1 + | M T 1 | + · · · + | M T d | . (1) (i) Fix i . If no facet of N T i in tersects N T [ u ], then ( M \ M T i ) ∪ N T i is a matc hing of N ( T ), hence it has cardinality at most ν . Since | M \ M T i | = ν − | M T i | , this giv es | N T i | ≤ | M T i | . Similarly , ( N \ N T i ) ∪ M T i is a matc hing, so | M T i | ≤ | N T i | . Therefore | M T i | = | N T i | . (ii) If N T [ u ] / ∈ N and u ∈ V N , then there exists a vertex v adjacent to u such that N T [ v ] ∈ N , and N = { N T [ v ] } ⊔ N T 1 ⊔ · · · ⊔ N T d , ν = 1 + | N T 1 | + · · · + | N T d | . (2) W e may assume that v ∈ V ( T 1 ). F or eac h i = 2 , . . . , d , the facets in M T i are disjoin t from N T [ v ]. Hence, if | N T i | = | M T i | − 1 for some i  = 1, then replacing the facets of N T i in N b y those of M T i pro duces a matc hing of N ( T ) of size ν + 1, a con tradiction. On the other hand, if | N T 1 | = | M T 1 | + 1, then replacing the facets of M T 1 in M by those of N T 1 yields a matc hing of size ν + 1, since all facets of N T 1 are disjoin t from N T [ u ], again a contradiction. Therefore, by Lemma 3.8, | M T 1 | − 1 ≤ | N T 1 | ≤ | M T 1 | , | M T i | ≤ | N T i | ≤ | M T i | + 1 for i = 2 , . . . , d. Using (1) and (2), this implies that either | N T i | = | M T i | for all i , or w e hav e | N T 1 | = | M T 1 | − 1, | N T j | = | M T j | + 1 for some j > 1, and | N T k | = | M T k | for all k  = 1 , j . In addition, it follows from (i) that N T j in tersects N T [ u ]. (iii) If u / ∈ V N , then N = N T 1 ⊔ · · · ⊔ N T d , ν = | N T 1 | + · · · + | N T d | . (3) 8 Supp ose that | N T i | = | M T i | − 1 for some i . Replacing the facets of N T i in N b y those of M T i pro duces a matching of N ( T ) of size ν + 1, again a con tradiction. Therefore, by Lemma 3.8, w e ha ve | M T i | ≤ | N T i | ≤ | M T i | + 1 for all i . Finally , using (1) and (3), it follows that there exists exactly one index i suc h that | N T i | = | M T i | + 1, and from (i) w e hav e that N T i in tersects N T [ u ]. □ Lemma 3.10. L et T b e a tr e e and let M and N b e two ν -matchings of N ( T ) . L et F ∈ M and supp ose that |{ G ∈ N : F ∩ G  = ∅}| = s > 2 . Then ther e exists a ν -matching W of N ( T ) diﬀer ent fr om M and N such that W ⊂ M ∪ N with |{ H ∈ W : F ∩ H  = ∅}| = 2 . Pr o of. Let F = N T [ u ] ∈ M . Then deg T ( u ) = d ≥ s . Since s > 3, we hav e N T [ u ] / ∈ N . Let T 1 , . . . , T d b e the connected comp onen ts of T \ { u } . By Lemma 3.9(i)–(iii), the inequality | N T i |  = | M T i | can o ccur for at most tw o indices i , and only for those i suc h that N T i con tains a facet in tersecting F . Since s > 2, there exist at least t w o indices i  = j such that N T i and N T j eac h contain a facet intersecting F . After relab eling, w e ma y assume that F in tersects a facet in eac h of N T 1 and N T 2 and that | N T i | = | M T i | for all i = 3 , . . . , d . Deﬁne W = N T 1 ⊔ N T 2 ⊔ M T 3 ⊔ · · · ⊔ M T d . Since all facets in v olv ed lie in pairwise disjoint comp onents of T \ { u } , the set W is a ν -matc hing of N ( T ). Moreov er, F do es not intersect any facet of M T 3 , . . . , M T d , and hence |{ H ∈ W : F ∩ H  = ∅}| = 2 as required. □ 4. Componentwise linearity of highest non-v an ishing squarefree po wer for trees In this section, we establish the necessary and suﬃcien t condition for the comp onent wise lin- earit y of highest non-v anishing squarefree pow er of the closed neigh b orho o d ideal of a tree. W e introduce the following t w o conditions on a tree T : (C1) T do es not contain a path r 1 , r , s, t, t 1 with deg T ( s ) ≥ 3 such that the tw o distinct ν - matc hings { N T [ r ] } ∪ Y and { N T [ t ] } ∪ Y exist, where Y is a ( ν − 1)-matc hing of N ( T ). (C2) T do es not con tain a path p 1 = r, p 2 , . . . , p 3 n − 1 = s , for n ≥ 1, with deg T ( r ) ≥ 3 and deg T ( s ) ≥ 3, suc h that the sets { N T [ p 1 ] , N T [ p 4 ] , . . . , N T [ p 3 n − 2 ] } ∪ Y and { N T [ p 2 ] , N T [ p 5 ] , . . . , N T [ p 3 n − 1 ] } ∪ Y eac h form a ν -matching of N ( T ), where Y is a ( ν − n )-matching of N ( T ). The following theorems show that Conditions (C1) and (C2) are necessary for the ν -th square- free p ow er of the closed neigh b orho o d ideal of a tree to b e comp onen t wise linear. 9 Theorem 4.1. L et T b e a tr e e that do es not satisfy Condition (C1) , and let J = N I ( T ) . Then J [ ν ] is not c omp onentwise line ar. Pr o of. Assume that T contains a path with edges { r 1 , r } , { r , s } , { s, t } , { t, t 1 } such that deg T ( s ) ≥ 3 and the monomials x N T [ r ] Y and x N T [ t ] Y b elong to G ( J [ ν ] ), where Y ∈ G ( J [ ν − 1] ). Thus T do es not satisfy Condition (C1). W rite N T [ r ] = { r , s, r 1 , r 2 , . . . , r d } and N T [ t ] = { t, s, t 1 , t 2 , . . . , t d ′ } , where d, d ′ ≥ 1. Set u = x A ( x N T [ r ] Y ) and v = x B ( x N T [ t ] Y ) , where A = { t 2 , . . . , t d ′ } and B = { r 2 , . . . , r d } , so that deg ( u ) = deg ( v ) = a . Let I := J [ ν ] . Then u, v ∈ I ⟨ a ⟩ . By Theorem 2.7, it suﬃces to show that u and v are disconnected in the graph G ( u,v ) I ⟨ a ⟩ . Let U = { F , F 2 . . . , F ν } and U ′ = { G, F 2 . . . , F ν } b e the ν -matc hing corresponding to u and v resp ectiv ely where F = N T [ r ], G = N T [ t ] and supp( Y ) = S ν i =2 F i . By the construction of u and v , w e ha v e supp( v ) \ supp( u ) = { t, t 1 } and supp( u ) \ supp( v ) = { r , r 1 } , and therefore supp(lcm( u, v )) = supp( u ) ∪ { t, t 1 } = supp( v ) ∪ { r , r 1 } = V U ∪ N T [ t ] = V U ′ ∪ N T [ r ] (4) Let w ∈ G ( u,v ) I ⟨ a ⟩ and W = { H 1 , . . . , H ν } b e the ν -matc hing corresp onding to w . Remo ving the edge { s, t } from T yields tw o connected comp onents T 1 and T 2 , where T 1 con tains s and T 2 con tains t . Note that N T [ s ] and N T [ t ] are the only facets of N ( T ) not entirely con tained in either V ( T 1 ) or V ( T 2 ). Hence, if N T [ s ] , N T [ t ] / ∈ W, then | U T 1 | = | W T 1 | and | U T 2 | = | W T 2 | . (5) Claim . W e hav e N T [ s ] / ∈ W . Pr o of of claim: Supp ose that N T [ s ] ∈ W . Then k = | U T 1 | = | W T 1 ∪ { N T [ s ] }| , that is U T 1 and W ′ = W T 1 ∪ { N T [ s ] } gives a k -matc hing of T . W e hav e N T [ r ] ∈ U T 1 . F ollowing the Construction 3.1, consider the bipartite graph B ( U T 1 ,W T 1 ) on 2 k ′ v ertices where 2 k ′ = 2 k − 2 t with t = | U T 1 ∩ W T 1 | . Then { N T [ r ] , N T [ s ] } forms an edge in B ( U T 1 ,W T 1 ) b ecause s ∈ N T [ r ] ∩ N T [ s ]. Due to our assumption deg T ( s ) ≥ 3, there exists z ∈ N T [ s ] \ { r, s, t } ⊂ V W . Since V W ′ ⊂ V U T 1 ∪ { t } due to (4), it follo ws that z ∈ V U T 1 and there exists a facet F ∈ U T 1 with F  = N T 1 [ r ] such that z ∈ F . Hence, the degree of N T 1 [ s ] in B ( U T 1 ,W T 1 ) is at least 2. Moreov er, since V W T 1 \ N T 1 [ s ] ⊂ V U T 1 , it follo ws from Remark 3.2 that the degree of eac h vertex in partition set W T 1 of B ( U T 1 ,W T 1 ) is also at least 2. This sho ws that B ( U T 1 ,W T 1 ) has more than 2 k ′ edges, and it is not tree, a contradiction to Lemma 3.3. This prov es our claim. Since deg(lcm( u, v )) = a + 2, it follows that { u, v } is not an edge in G ( u,v ) I ⟨ a ⟩ . In fact, given any w ∈ G ( u,v ) I ⟨ a ⟩ , if N T [ r ] ⊂ supp( w ), then w is not adjacen t to v . Then to prov e that u and v are disconnected in G ( u,v ) I ⟨ a ⟩ , it is enough to pro v e that if a monomial w ∈ G ( u,v ) I ⟨ a ⟩ is connected to u , then N T [ r ] ∈ W . W e prov e this b y applying induction on d ( u, w ), that is, the length of the shortest path in G ( u,v ) I ⟨ a ⟩ connecting u and v . Note that the assertion trivially holds for d ( u, w ) = 0 b ecause in this case u = w . No w assume that assertion is true for all d ( u, w ) ≤ k − 1 where k ≥ 1. No w, let w ∈ G ( u,v ) I ⟨ a ⟩ b e such that d ( u, w ) = k . 10 Since w ∈ G ( u,v ) I ⟨ a ⟩ , we ha v e that deg(lcm( u, w )) ≤ a + 2. First consider the case when deg(lcm( u, w )) = a + 1. Then u and w diﬀer in exactly one v ariable, that is, there exists some x ∈ supp( u ) suc h that w = ( u/x ) y where y ∈ { t, t 1 } , due to (4). This sho ws that N T [ t ] ⊂ supp( w ) b ecause t or t 1 is missing from supp( w ). In particular, N T [ t ] / ∈ W , and from (5) it follows that | U T 1 | = | W T 1 | . On the other hand, w e ha v e V W T 1 ⊂ V U T 1 due to (4). Then by Prop osition 3.4, we conclude that U T 1 = W T 1 , and hence N T 1 [ r ] ∈ W and N T 1 [ r ] ⊂ supp( w ), as required. This also sho ws that x is a vertex of T 2 . No w, let deg(lcm( u, w )) = a + 2. Then u and w diﬀer in exactly tw o v ariables, that is, there exists some x, z ∈ supp( u ) such that w = ( u/xz ) tt 1 , due to (4). Since w is connected to u , there exists a monomial w ′ ∈ G ( u,v ) I ⟨ a ⟩ suc h that w is adjacent to w ′ and d ( u, w ′ ) = k − 1, and b y the induction hypothesis N T 1 [ r ] is con tained in the matc hing corresp onding to w ′ , and w ′ = ( u/x ′ ) y , where y ∈ { t, t 1 } and x ′ is in T 2 . If N T [ t ] / ∈ W , then again, as argued ab o v e, we obtain the desired conclusion. T o complete the pro of, it only remains to rule out the case N T [ t ] ∈ W . Assume that N T [ t ] ∈ W . If b oth x and z b elong to T 1 , then deg (lcm( v , w ′ )) = a + 2 and w cannot b e adjacen t to w ′ . So, at least one of the v ertex, sa y x b elongs to T 2 . Since N T [ t ] ∈ U ′ , w e obtain | U ′ T 2 | = | W T 2 | . Since W T 2 ⊆ U ′ T 2 due to (4), by Prop osition 3.4, we conclude that U ′ T 2 = W T 2 . But x ∈ V U ′ T 2 , w e obtain a con tradiction to x / ∈ supp( w ). □ Theorem 4.2. L et T b e a tr e e that do es not satisfy Condition (C2) , and let J = N I ( T ) . Then J [ ν ] is not c omp onentwise line ar. Pr o of. Let r and s b e t w o v ertices of T with deg ( r ) ≥ deg ( s ) ≥ 3. Assume that P : p 1 , . . . , p 3 n − 1 is a path on 3 n − 1 vertices connecting r and s , that is, p 1 = r and p 3 n − 1 = s . F urthermore, assume that there exist monomials u ′ = x N T [ p 1 ] x N T [ p 4 ] · · · x N T [ p 3 n − 2 ] Y and v ′ = x N T [ p 2 ] x N T [ p 5 ] · · · x N T [ p 3 n − 1 ] Y b elong to G ( J [ ν ] ), where Y ∈ J [ ν − n ] . Thus T do es not satisfy Condition (C2). Let P b e a shortest path that satisﬁes the ab ov e condition. First, we claim that if n ≥ 2 then one has deg T ( p i ) = 2 for all i ≡ 1 , 2 (mo d 3) with i ∈ { 2 , . . . , 3 n − 2 }} . Supp ose, to the con trary , that deg T ( p i ) ≥ 3 for some i ∈ { 2 , . . . , 3 n − 2 } with i ≡ 2 (mo d 3). In this case, w e hav e u 1 = x N T [ p 1 ] x N T [ p 4 ] · · · x N T [ p i − 1 ] Z and v 1 = x N T [ p 2 ] x N T [ p 5 ] · · · x N T [ p i ] Z b elongs to G ( J [ ν ] ), where Z ∈ J [ ν − m ] . This con tradicts our assumption that P is a shortest path that satisﬁes the h yp othesis. Similarly , the case deg T ( p i ) ≥ 3 for some i ∈ { 2 , . . . , 3 n − 2 } with i ≡ 1 (mo d 3) is not p ossible. Hence, the claim follo ws. Let N T [ r ] = { r , r 1 , . . . , r d } and N T [ s ] = { s, s 1 , . . . , s d ′ } . If n = 1, set r 1 = s and s 1 = r ; if n ≥ 2, c ho ose r 1 , s 1 ∈ V P , that is, r 1 = p 1 and s 1 = p 3 n − 2 . Set u = x A ( x N T [ r ] Y ) and v = x B ( x N T [ t ] Y ) , where A = { s 4 , . . . , s d ′ } and B = { r 4 , . . . , r d } , so that deg ( u ) = deg( v ) = a . Let I := J [ ν ] . Then u, v ∈ I ⟨ a ⟩ . By Theorem 2.7, it suﬃces to show that u and v are disconnected in the graph G ( u,v ) I ⟨ a ⟩ . Let U and U ′ b e the ν -matc hing corresp onding to u and v . By the construction of u and v , w e hav e supp( v ) \ supp( u ) = { s 2 , s 3 } and supp( u ) \ supp( v ) = { r 2 , r 3 } , and therefore supp(lcm( u, v )) = supp( u ) ∪ { s 2 , s 3 } = supp( v ) ∪ { r 2 , r 3 } = V U ∪ N T [ s ] = V U ′ ∪ N T [ r ] . (6) Let T 1 , T 2 , . . . , T d ′ b e the conncted comp onents of T \ { s } such that s i lies in T i for each i = 1 , . . . , d ′ . Let w ∈ G ( u,v ) I ⟨ a ⟩ and W = { H 1 , . . . , H ν } b e the ν -matc hing corresp onding to w . 11 Since U and W are maximal matching of T , we note that if N T [ s ] , N T [ s i ] / ∈ W, then | U T 1 ∪{ s } | = | W T 1 ∪{ s } | (7) Claim: N T [ s i ] / ∈ W for all i = 2 , 4 , 5 , . . . , d ′ . Pro of of claim: Assume that N T [ s i ] ∈ W for i = 2. The same argumen t applies for any other c hoice of i . Let F = N T [ s 1 ] and G = N T [ s 2 ]. Let M = U T 1 ∪ U T 2 ∪ { F } ⊆ U and M ′ = W T 1 ∪ W T 2 ∪ { G } ⊆ W . Then | M | = | M ′ | b ecause U and W are maximal matc hing of T . F ollowing the Construction 3.1, consider the bipartite graph B ( M ,M ′ ) on 2 k ′ v ertices where 2 k ′ = 2 | M | − 2 k with k = | M ∩ M ′ | . Then { F , G } forms an edge in B ( M ,M ′ ) , since s ∈ F ∩ G . Since G ∈ M ′ , there exists z ∈ G \ { s, s 2 } ⊂ V N . Since V M ′ \ A ⊂ V M ∪ { s 2 } , it follo ws that z ∈ V M and there exists a facet F ′ ∈ M with F ′  = F such that z ∈ F ′ . Hence, the degree of G in B ( M ,M ′ ) is at least 2. Moreov er, since V M ′ \ { G } ⊆ V M , it follo ws from Remark 3.2 that in B ( M ,M ′ ) the degree of each G ′ ∈ M ′ is also at least 2. This sho ws that B ( M ,M ′ ) has more than 2 k ′ edges, and it is not tree, a con tradiction to Lemma 3.3. This prov es our claim. W e pro ceed as in the pro of of Theorem 4.1. Since deg(lcm( u, v )) = a + 2, it follows that { u, v } is not an edge in G ( u,v ) I ⟨ a ⟩ . In fact, giv en any w ∈ G ( u,v ) I ⟨ a ⟩ , if N T [ r ] ⊂ supp( w ), then w is not adjacen t to v . Then to prov e that u and v are disconnected in G ( u,v ) I ⟨ a ⟩ , it is enough to pro v e that if a monomial w ∈ G ( u,v ) I ⟨ a ⟩ is connected to u , then N T [ r ] ⊂ W , the matching corresp onding to w . W e prov e this b y applying induction on d ( u, w ). Note that the assertion trivially holds for d ( u, w ) = 0 b ecause in this case u = w . Now assume that assertion is true for all d ( u, w ) ≤ k − 1 where k ≥ 1. Now, let w ∈ G ( u,v ) I ⟨ a ⟩ b e such that d ( u, w ) = k . Since w ∈ G ( u,v ) I ⟨ a ⟩ , we hav e that deg (lcm( u, w )) ≤ a + 2. First, consider the case when deg(lcm( u, w )) = a + 1. Then, there exists some x ∈ supp( u ) such that w = ( u/x ) y where y ∈ { s 2 , s 3 } , due to (6). This shows that N T [ s ] ⊂ supp( w ) b ecause s 2 or s 3 is missing from supp( w ). In particular, N T [ s ] / ∈ W , and from (7) it follo ws that | U T 1 ∪{ s } | = | W T 1 ∪{ s } | . On the other hand, w e ha ve V W T 1 ⊂ V U T 1 due to (6). Then b y Prop osition 3.4, we conclude that U T 1 ∪{ s } = W T 1 ∪{ s } , and hence N T 1 [ r ] ∈ W , as required. This also sho ws that x is a vertex of T 2 . No w, let deg(lcm( u, w )) = a + 2. Then u and w diﬀer in exactly tw o v ariables, that is, there exists some x, z ∈ supp( u ) such that w = ( u/xz ) s 2 s 3 , due to (6). Since w is connected to u , there exists a monomial w ′ ∈ G ( u,v ) I ⟨ a ⟩ suc h that w is adjacent to w ′ and d ( u, w ′ ) = k − 1, and b y the induction hypothesis N T 1 [ r ] is con tained in the matc hing corresp onding to w ′ , and w ′ = ( u/x ′ ) y , where y ∈ { s 2 , s 3 } and x ′ is in T 2 . If N T [ s ] / ∈ W , then again, as argued ab ov e, w e obtain the desired conclusion. T o complete the pro of, it only remains to rule out the case N T [ s ] ∈ W . Assume that N T [ s ] ∈ W . Since N T [ s ] ∈ U ′ , w e obtain | U ′ T 2 | = | W T 2 | . W e also hav e W T 2 ⊆ U ′ T 2 due to (6), by Prop osition 3.4, we conclude that U ′ T 2 = W T 2 . This shows that both x and z b elong to T 1 . This gives deg(lcm( v , w ′ )) = a + 2 and w cannot b e adjacen t to w ′ , a con tradiction. □ Next, w e show that (C1) and (C2) are necessary for comp onen t wise linearit y of highest non- v anishing squarefree p o w er of N I ( T ). In fact, w e sho w that if T satisﬁes (C1) and (C2), then N I ( T ) [ ν ] admits linear quotien t. T o do this, we now deﬁne tw o total orders on the ν -matchings of N ( T ), namely > lex and > ℓ . 12 Notation 4.3. Let T b e a tree and let x b e a p enden t v ertex of T . W e view T as a ro oted tree with root x . F or a vertex u ∈ V ( T ), w e deﬁne lev el( u ) to b e the distance from x to u . W e assign the following notation to the facets of N ( T ). F or a vertex u ∈ V ( T ), w e write F i,j,k = N T [ u ] if | N T [ u ] | = i , level( u ) = j , and u is the k -th v ertex in a ﬁxed ordering of the vertices at level j (see Remark 4.4). W e deﬁne a total order on F ( N ( T )) by setting F i,j,k > F i ′ ,j ′ ,k ′ if one of the follo wing holds: (i) i < i ′ ; (ii) i = i ′ and j < j ′ ; (iii) i = i ′ , j = j ′ , and k < k ′ . Let M = { F i 1 ,j 1 ,k 1 , . . . , F i ν ,j ν ,k ν } and N = { F i ′ 1 ,j ′ 1 ,k ′ 1 , . . . , F i ′ ν ,j ′ ν ,k ′ ν } b e t w o ν -matc hings of N ( T ), written in decreasing order with resp ect to the ab ov e facet order. Let s = min { ℓ : F i ℓ ,j ℓ ,k ℓ  = F i ′ ℓ ,j ′ ℓ ,k ′ ℓ } . W e deﬁne a lexicographic order on ν -matc hings b y setting M > lex N if F i s ,j s ,k s > F i ′ s ,j ′ s ,k ′ s . By Corollary 3.5, each generator of N I ( T ) [ ν ] corresp onds uniquely to a ν -matching of N ( T ), and this induces a total order > lex on the minimal generating set G ( N I ( T ) [ ν ] ). Remark 4.4. The index k is used only to imp ose a total order on facets corresponding to v ertices with the same v alue of level( u ) and with neighborho o ds of the same size; any ﬁxed choice of k suﬃces. Example 4.5. Let T b e the tree on [14] sho wn in Figure 1, ro oted at the p enden t v ertex 1. Using Notation 4.3, we lab el the facets of the closed neigh b orho o d complex N ( T ) as follo ws: F 2 , 0 , 1 = N G [1] = { 1 , 2 } , F 2 , 6 , 1 = N G [14] = { 13 , 14 } , F 2 , 10 , 1 = N G [11] = { 10 , 11 } , F 3 , 2 , 1 = N G [3] = { 2 , 3 , 4 } , F 3 , 4 , 1 = N G [12] = { 4 , 12 , 13 } , F 3 , 4 , 2 = N G [5] = { 4 , 5 , 6 } , F 3 , 5 , 2 = N G [6] = { 5 , 6 , 7 } , F 3 , 6 , 2 = N G [7] = { 6 , 7 , 8 } , F 3 , 7 , 1 = N G [8] = { 7 , 8 , 9 } , F 3 , 8 , 1 = N G [9] = { 8 , 9 , 10 } , F 4 , 3 , 1 = N G [4] = { 3 , 4 , 5 , 12 } . According to the total order on facets deﬁned in Notation 4.3, we obtain F 2 , 0 , 1 > F 2 , 6 , 1 > F 2 , 10 , 1 > F 3 , 2 , 1 > F 3 , 4 , 1 > F 3 , 4 , 2 > F 3 , 5 , 2 > F 3 , 6 , 2 > F 3 , 7 , 1 > F 3 , 8 , 1 > F 4 , 3 , 1 . The 5-matchings of N ( T ) are M 1 = { F 2 , 0 , 1 , F 2 , 6 , 1 , F 2 , 10 , 1 , F 3 , 4 , 2 , F 3 , 7 , 1 } , M 2 = { F 2 , 0 , 1 , F 2 , 6 , 1 , F 2 , 10 , 1 , F 3 , 6 , 2 , F 4 , 3 , 1 } , M 3 = { F 2 , 0 , 1 , F 2 , 6 , 1 , F 2 , 10 , 1 , F 3 , 7 , 1 , F 4 , 3 , 1 } . With resp ect to the lexicographic order on ν -matchings introduced in Notation 4.3, we ha v e M 1 > lex M 2 > lex M 3 . Let U b e the unique ν -matc hing which is maximal with resp ect to > lex (Notation 4.3). F or an y ν -matc hing M of N ( T ), we deﬁne the level of M with resp ect to U , denoted b y level U ( M ), b y lev el U ( M ) := ν − | U ∩ M | . F urthermore, for a ν -matching M , we denote by β ( M ) :=   { F ∈ M : | F | = 2 }   the num ber of facets of M of cardinality t w o. 13 1 2 3 4 10 9 8 7 6 5 11 12 13 14 Figure 1. The tree T . Notation 4.6. W e deﬁne a total order > ℓ on the set of ν -matc hings of N ( T ) as follo ws. F or tw o ν -matc hings M and N , we set M > ℓ N if one of the follo wing holds: (i) β ( M ) > β ( N ); (ii) β ( M ) = β ( N ) and level U ( M ) < level U ( N ); (iii) β ( M ) = β ( N ), level U ( M ) = lev el U ( N ), and M > lex N (see Notation 4.3). Via the bijection betw een ν -matchings of N ( T ) and the minimal generators of N I ( T ) [ ν ] (Corol- lary 3.5), this order induces a total order on G ( N I ( T ) [ ν ] ), which w e also denote by > ℓ . Example 4.7. Let T b e the tree from Example 4.5, and let M 1 , M 2 , M 3 b e the ν -matchings of N ( T ) describ ed there. Then M 1 is the maximal matching with resp ect to > lex and w e hav e lev el U ( M 3 ) = 1 and lev el U ( M 2 ) = 2 . With resp ect to the total order > ℓ deﬁned in Notation 4.6, w e obtain M 1 > ℓ M 3 > ℓ M 2 . On the other hand, with resp ect to the lexicographic order > lex (see Example 4.5), we ha ve M 1 > lex M 2 > lex M 3 . This example illustrates that the order > ℓ reﬁnes the lexicographic order b y incorp orating b oth the num ber of size-tw o facets and the n um b er of common facets with U . In the subsequen t results, we describ e the structural prop erties of ν -matc hings of N ( T ) that satisﬁes (C1) and (C2). Remark 4.8. Let T b e a tree satisfying (C1). Let M and N b e tw o ν -matchings of N ( T ). Supp ose that N T [ r ] ∈ M and N T [ t ] ∈ N satisfy N T [ r ] ∩ N T [ t ] = { s } and deg T ( r ) , deg T ( t ) ≥ 2. (1) Let T 1 , . . . , T d b e the connected comp onen ts of T \ { r } , and assume that t ∈ T 1 . If | N T 1 | = | M T 1 | + 1, then deﬁne Y =  M \ M T 1  ∪  N T 1 \ { N T [ t ] }  . Then Y is a ( ν − 1)-matching of N ( T ), and b oth { N T [ r ] } ∪ Y and { N T [ t ] } ∪ Y are ν -matc hings of N ( T ). Hence, b y (C1), it follows that deg T ( s ) = 2. (2) If N T [ r ] do es not intersect any other facet of N . Then Y = ( N \ N T [ t ]) is a ( ν − 1)- matc hing of N ( T ), and b oth { N T [ r ] } ∪ Y and { N T [ t ] } ∪ Y = N are ν -matc hings of N ( T ). Hence, by (C1), it follo ws that deg T ( s ) = 2. Prop osition 4.9. L et T b e a tr e e satisfying (C1) and (C2) . L et U b e the unique ν -matching that is maximal with r esp e ct to > lex (Notation 4.3). Then every fac et F ∈ U with | F | ≥ 4 b elongs to every ν -matching N of N ( T ) . 14 Pr o of. Let F = N T [ i 1 ] ∈ U , where deg T ( i 1 ) = d ≥ 3. Let T 1 , . . . , T d b e the connected components of T \ { i 1 } . Supp ose, for a contradiction, that there exists a ν -matc hing N such that N T [ i 1 ] / ∈ N . W e consider tw o cases: (1) i 1 / ∈ V N , (2) i 1 ∈ V N . Case I. Supp ose that i 1 / ∈ V N . By Lemma 3.9, there exists j ∈ [ d ], say j = 1, suc h that | N T 1 | = | U T 1 | + 1, and a facet G ∈ N T 1 with G ∩ N T [ i 1 ]  = ∅ . W rite G = N T [ i 3 ] and assume N T [ i 1 ] ∩ G = { i 2 } , where i 1 , i 2 , i 3 form a path in T . By Remark 4.8(1), we ha ve deg T ( i 2 ) = 2. · · · i 1 i 2 i 4 i 3 i 5 i 6 . . . . . . · · · · · · · · · Figure 2. The neighborho o ds of the red v ertices b elong to V U , while the neigh- b orho o ds of the blue vertices b elong to V N . If i 3 / ∈ V U , then U ′ = ( U \ N T [ i 1 ]) ∪ N T [ i 2 ] is a ν -matching of T . Since N T [ i 2 ] > N T [ i 1 ] with resp ect to the order in Notation 4.3, w e obtain U ′ > lex U and this con tradicts the maximality of U . Hence i 3 ∈ V U , and there exists a neigh bor i 4 of i 3 with N T [ i 4 ] ∈ U and N T [ i 2 ] ∩ N T [ i 4 ] = { i 3 } . Moreo v er, i 4 is not a leaf of T , since otherwise N T [ i 4 ] ⊊ N T [ i 3 ], contradicting that N T [ i 3 ] is a facet. Let S 1 , . . . , S t b e the connected comp onen ts of T \ { i 4 } , and assume that i 3 ∈ S 1 . Since | N T 1 | = | U T 1 | + 1, we ha v e | N S 1 | = | U S 1 | − 1. By Lemma 3.9(ii), there exists j ∈ [ t ], sa y j = 2, suc h that | N S 2 | = | U S 2 | + 1. F urthermore, by Lemma 3.9(i), there exists a facet N T [ i 6 ] ∈ N S 2 suc h that N T [ i 4 ] ∩ N T [ i 6 ] = { i 5 } (see Figure 2). By Remark 4.8(1), it follows that deg T ( i 5 ) = 2. If i 6 / ∈ V U , then the set U ′′ =  U \ { N T [ i 1 ] , N T [ i 4 ] }  ∪ { N T [ i 2 ] , N T [ i 5 ] } is a ν -matching of N ( T ). Since i 4 lies b et ween i 2 and i 5 in T and | N T [ i 2 ] | = | N T [ i 5 ] | = 3, it follows from Notation 4.3 that either N T [ i 2 ] > N T [ i 4 ] or N T [ i 5 ] > N T [ i 4 ]. Consequently , U ′′ > lex U , contradicting the maximality of U . Hence i 6 ∈ V U . Rep eating the same argument at i 6 , we extend the path i 1 , i 2 , i 3 , i 4 , i 5 , i 6 , . . . strictly farther from i 1 , where ev ery second v ertex has degree 2 and the terminal vertex lies in V U . Since T is ﬁnite, this pro cess must reach a leaf, where the construction cannot con tin ue. At that stage the alternativ e i 3 k / ∈ V U m ust o ccur, yielding a contradiction to the maximality of U . Therefore, the initial assumption N T [ i 1 ] / ∈ N is imp ossible. Case II. Supp ose that i 1 ∈ V N . Then there exists a v ertex i 2 adjacen t to i 1 suc h that N T [ i 2 ] ∈ N . Assume i 2 ∈ V ( T 1 ). By Lemma 3.9, one of the following occurs: (i) There exists j , sa y j = 2, suc h that | N T 2 | = | U T 2 | + 1. Since i 2 ∈ V ( T 1 ), it follo ws from the pro of of Lemma 3.9(ii) that | N T 1 | = | U T 1 | − 1, and | N T k | = | U T k | for all k  = 1 , 2. (ii) | N T j | = | U T j | for all j . Supp ose that (i) holds. Then M = ( U \ { N T [ i 1 ] } ) ∪ N T 2 is a ν -matc hing of N ( T ), since the comp onen twise cardinalities agree. Moreov er, N T [ i 1 ] / ∈ M and i 1 / ∈ V M . As | N T [ i 1 ] | ≥ 4, this con tradicts Case I, whic h shows that no such ν -matching can exist. Hence (i) do es not o ccur. Therefore | N T j | = | U T j | for all j . Set Y = N T 1 ∪ U T 2 ∪ · · · ∪ U T d . 15 Then Y is a ( ν − 1)-matching of N ( T ), and b oth Y ∪ N T [ i 1 ] and Y ∪ N T [ i 2 ] are ν -matchings of N ( T ). These tw o matc hings are distinct, since i 1  = i 2 . Because T satisﬁes (C2) and deg T ( i 1 ) ≥ 3, it follo ws that deg T ( i 2 ) = 2. Since Y ∪ N T [ i 2 ] satisﬁes the same assumptions as N in Case I I, w e ma y set N = Y ∪ N T [ i 2 ] in the rest of the pro of. Since N T [ i 2 ] > N T [ i 1 ] with resp ect to the order in Notation 4.3, the maximalit y of U implies that N T [ i 2 ] ∩ V U \{ N T [ i 1 ] }  = ∅ ; otherwise replacing N T [ i 1 ] b y N T [ i 2 ] w ould yield a ν -matching strictly larger than U , a contradiction. Therefore, there exists a facet N T [ i 4 ] ∈ U \ N T [ i 1 ] such that N T [ i 2 ] ∩ N T [ i 4 ] = { i 3 } . Supp ose that i 4 / ∈ V N . If deg T ( i 4 ) > 2, then Case I applied at i 4 yields N T [ i 4 ] ∈ N , a con tradiction; hence deg T ( i 4 ) ≤ 2. If N T [ i 4 ] intersects only N T [ i 2 ] among the facets of N , then replacing N T [ i 2 ] b y N T [ i 4 ] and N T [ i 1 ] produces a ( ν + 1)-matching, a contradiction. Thus N T [ i 4 ] m ust intersect another facet of N , and we obtain deg T ( i 4 ) = 2. Let i 5 b e the neigh b or of i 4 distinct from i 3 , and let N T [ i 6 ] ∈ N b e the facet in tersecting N T [ i 4 ], so that i 4 , i 5 , i 6 form a path in T . Then N T [ i 4 ] intersects only N T [ i 6 ] among the facets of W ′ = ( U \ U T 1 ) ∪ N T 1 . By Condition (C1), we obtain deg T ( i 5 ) = 2. Since i 4 / ∈ V N , replacing N T [ i 6 ] in N by N T [ i 5 ] yields a ν -matc hing N ′ satisfying the same assumptions as N in Case I I. Hence we may replace N by N ′ and contin ue; in particular, we ma y assume i 4 ∈ V N . No w assume that i 4 ∈ V N . Then there exists a vertex i 5 suc h that i 4 ∈ N T [ i 5 ] ∈ N . Note that i 5 cannot b e a leaf of T , since otherwise N T [ i 5 ] ⊊ N T [ i 4 ], contradicting the fact that N T [ i 4 ] is a facet of N ( T ). W e distinguish according to the degree of i 4 . If deg T ( i 4 ) ≥ 3, then the same argumen t used at the b eginning of Case I I (applied now to the pair i 4 , i 5 in place of i 1 , i 2 ) and Condition (C2) yields deg T ( i 5 ) = 2. If deg T ( i 4 ) = 2, then i 1 , . . . , i 5 form a path in T , and the set M = N \ { N T [ i 2 ] , N T [ i 5 ] } is a ( ν − 2)-matching such that b oth M ∪ { N T [ i 2 ] , N T [ i 5 ] } = N and M ∪ { N T [ i 1 ] , N T [ i 4 ] } are ν -matc hings. Since T satisﬁes (C2) and deg T ( i 1 ) ≥ 3, it follo ws again that deg T ( i 5 ) = 2. Th us in all cases deg T ( i 5 ) = 2. Because i 4 lies b etw een i 2 and i 5 in T , it follows from Notation 4.3 that either N T [ i 2 ] > N T [ i 4 ] or N T [ i 5 ] > N T [ i 4 ]. Consequently , ( U \ { N T [ i 1 ] , N T [ i 4 ] } ) ∪ { N T [ i 2 ] , N T [ i 5 ] } > lex U, con tradicting the maximalit y of U . i 1 i 2 i 4 i 3 i 5 i 6 i 7 . . . . . . . . . . . . . . . . . . Figure 3. The neighborho o ds of the red v ertices b elong to V U , while the neigh- b orho o ds of the blue vertices b elong to V N . Th us N T [ i 5 ] ∩ V U \{ N T [ i 4 ] }  = ∅ , so there exists a facet N T [ i 7 ] ∈ U intersecting N T [ i 5 ]. Re- p eating the same argument alternately with facets of N and U , we extend the simple path i 1 , i 2 , i 3 , i 4 , i 5 , i 6 , i 7 , . . . strictly farther a w a y from i 1 in T , while eac h newly created intermediate v ertex has degree 2. Since T is ﬁnite, this pro cess must terminate, and at the terminal step the alternative N T [ i 3 k +2 ] ∩ V U \{ N T [ i 3 k +1 ] }  = ∅ can no longer hold. Hence the complementary case must o ccur, which yields the con tradiction to the maximalit y of U . Therefore N T [ i 1 ] ∈ N , con tradicting our assumption. □ 16 Prop osition 4.10. L et T b e a tr e e satisfying (C1) and (C2) . L et U b e the unique ν -matching that is maximal with r esp e ct to > lex (Notation 4.3), and let N b e any ν -matching of N ( T ) . Supp ose that ther e exist fac ets F 3 ,a,b ∈ U and G 3 ,c,d ∈ N such that | F 3 ,a,b ∩ G 3 ,c,d | = 1 and F 3 ,a,b is disjoint fr om every other fac et of N . Then a < c , e quivalently F 3 ,a,b > lex G 3 ,c,d . Pr o of. Set F 3 ,a,b = N T [ i 1 ] and G 3 ,c,d = N T [ i 3 ] with F 3 ,a,b ∩ G 3 ,c,d = { i 2 } , where i 1 , i 2 , i 3 form a path in T . Since N T [ i 1 ] is disjoin t from all other facets of N , Remark 4.8(2) implies that deg T ( i 2 ) = 2. i 1 i 2 i 4 i 3 i 5 i 6 . . . . . . · · · Figure 4. The neighborho o ds of the red vertices b elong to V U and the the neigh- b orho o ds of the blue vertices b elong to V N . F ollowing Notation 4.3, w e ha v e N T [ i 2 ] = F 3 ,j,k . Assume for contradiction that c < a . Then c < j < a , equiv alently , N T [ i 3 ] > N T [ i 2 ] > N T [ i 1 ] . W e ﬁrst note that i 3 ∈ V U . Indeed, if i 3 / ∈ V U , then replacing N T [ i 1 ] in U by N T [ i 2 ] yields a ν -matc hing strictly larger than U , con tradicting its maximality . Hence i 3 ∈ V U , and there exists a neighbor i 4 of i 3 with N T [ i 4 ] ∈ U suc h that N T [ i 2 ] ∩ N T [ i 4 ] = { i 3 } . The v ertex i 4 is not a leaf of T , since otherwise N T [ i 4 ] ⊊ N T [ i 3 ], con tradicting that N T [ i 3 ] is a facet of N ( T ). Moreov er, b y Prop osition 4.9, we ha v e | N T [ i 4 ] | ≤ 3, and therefore | N T [ i 4 ] | = 3. F urthermore, N T [ i 4 ] m ust in tersect at least t w o facets of N ; otherwise replacing N T [ i 3 ] in N by N T [ i 1 ] and N T [ i 4 ] would pro duce a ( ν + 1)-matching, a con tradiction. Th us there exists a path i 4 , i 5 , i 6 in T suc h that N T [ i 4 ] ∩ N T [ i 6 ] = { i 5 } . In this situation the comp onent cardinalities diﬀer b y one in the comp onent con taining i 6 , so the hypotheses of Remark 4.8 are satisﬁed. Hence deg T ( i 5 ) = 2. W e next show that i 6 ∈ V U . If i 6 / ∈ V U , then replacing N T [ i 1 ] and N T [ i 4 ] in U by N T [ i 2 ] and N T [ i 5 ] pro duces a ν -matching strictly larger than U with resp ect to > lex , contradicting the maximalit y of U . Hence i 6 ∈ V U , so there exists a neigh b or i 7 of i 6 with N T [ i 7 ] ∈ U . As in the case of i 4 , the vertex i 7 is not a leaf of T , and we obtain a path i 6 , i 7 , i 8 in T . Consider the matc hing M = ( N \ { N T [ i 3 ] , N T [ i 6 ] } ) ∪ { N T [ i 2 ] , N T [ i 5 ] } . In this situation the component cardinalities again diﬀer by one in the comp onent con taining i 7 , so the h yp otheses of Remark 4.8 are satisﬁed. Hence deg T ( i 6 ) = 2. Rep eating the same argumen t alternately with facets of N and U , w e extend the path i 1 , i 2 , i 3 , i 4 , i 5 , i 6 , i 7 , . . . strictly farther aw a y from i 1 in T . A t each step the newly introduced v ertices hav e degree 2, while the terminal vertex lies in V U . Since T is ﬁnite, the pro cess must even tually terminate. A t the ﬁnal step the condition i 3 k ∈ V U fails, and the complemen tary case yields a con tradiction to the maximality of U . Hence a < c . □ Lemma 4.11. L et T b e a tr e e. L et U b e the unique ν -matching which is maximal with r esp e ct to > lex . L et M and N b e two ν -matchings with F ∈ M \ U and G ∈ N \ U . Assume that ( F \ G ) ∩ V N = ∅ . Then ther e exist H ∈ U such that H ∩ F  = ∅ and H ∩ G  = ∅ . 17 Pr o of. W e hav e F ∩ G  = ∅ because M and N are maximal matc hings. Let F = N T [ r ] and G = N T [ t ]. Supp ose, to the con trary , that for ev ery H ∈ U either H ∩ N T [ r ] = ∅ or H ∩ N T [ t ] = ∅ . Case I: Let | N T [ r ] ∩ N T [ t ] | = 1, and set N T [ r ] ∩ N T [ t ] = { s } . F rom our assumption we ha v e that N T [ x ] / ∈ U for all x ∈ N T [ s ]. Let T 1 , . . . , T d b e the connected comp onents of T \ r . Set s ∈ T 1 . By Lemma 3.9 and the assumption ( F \ G ) ∩ V N = ∅ , it follo ws that | N T 1 | = | M T 1 | + 1 and | N T i | = | M T i | for all i = 2 , . . . , d . Note that if | U T i | = | M T i | + 1 for some 2 ≤ i ≤ d , then U T i ∪ N T 1 ∪ d [ k =2 ,k  = i M T k forms a ( ν + 1)-matching of N ( T ), since eac h T i has pairwise disjoin t set of v ertices. This con tradicts the matching num b er of N ( T ). F urther, since N T [ x ] / ∈ U for all x ∈ N T [ s ], b y Lemma 3.9(i) we ha v e | U T 1 | = | M T 1 | . Then from Lemma 3.9, we obtain | U T i | = | M T i | for all i = 1 , . . . , d and N T [ y ] ∈ U for some y ∈ N T [ r ] \ { r , s } . Then replacing the facets of U T 1 in U by those of N T 1 pro duces a ( ν + 1)-matching, a contradiction. This shows that | N T [ r ] ∩ N T [ t ] | = 1 do es not hold. Case I I. Let | N T [ r ] ∩ N T [ t ] | = 2, where { r, t } ∈ E ( T ). Let T 1 , . . . , T d b e the connected comp onen ts of T \ r . Set t ∈ T 1 . F rom our assumption we ha v e that N T [ x ] / ∈ U for all x ∈ V N T [ r ] , N T [ t ] . In particular, r / ∈ V U . By Lemma 3.9 and the assumption ( F \ G ) ∩ V N = ∅ , it follows that | N T i | = | M T i | for all i = 1 , . . . , d . Note that if | U T i | = | M T i | + 1 for some 2 ≤ i ≤ d then U T i ∪ ( N \ N T i ) forms a ( ν + 1)-matching of N ( T ), since r / ∈ V U . This contradicts the matching n um b er of N ( T ). F urther, since N T [ x ] / ∈ U for all x ∈ N T [ t ], by Lemma 3.9(i) we ha v e | U T 1 | = | M T 1 | . Then from Lemma 3.9, w e obtain | U T i | = | M T i | for all i = 1 , . . . , d and r ∈ V U , a con tradiction. □ Prop osition 4.12. L et T b e a tr e e satisfying (C1) and (C2) . L et U b e the unique ν -matching maximal with r esp e ct to > lex (Notation 4.3), and let M and N b e two ν -matchings of N ( T ) such that: (i) M > ℓ N (Notation 4.6), (ii) ther e do not exist fac ets F ∈ M and G ∈ N with F > G such that F \ G is a singleton and F \ G / ∈ V N . Then ther e exist fac ets H ∈ U and G ∈ N , and a vertex x ∈ H \ G , such that W = ( N \ G ) ∪ H is a ν -matching and x ∈ V M \ V N . Pr o of. Let M and N b e t w o ν -matchings of N ( T ) satisfying (i) and (ii). W e ﬁrst normalize M as follows. Supp ose there exist F ∈ M \ U and G ∈ N with G > F such that ¯ M = ( M \ F ) ∪ { G } is a ν -matc hing. Then ¯ M > ℓ N still satisﬁes (i) and (ii). Hence it suﬃces to prov e the statement for ¯ M and N , since V ¯ M \ V N ⊆ V M \ V N . Consequently w e may assume: Assumption on M : F or ev ery F ∈ M \ U there is no G ∈ N with G > F such that ( M \ F ) ∪ { G } is a ν -matc hing. 18 Note that this assumption implies all one-dimensional facets of M and N coincide. (8) Indeed, let F ∈ M with | F | = 2. Then F intersects some G ∈ N , since otherwise N ∪ { F } w ould b e a ( ν + 1)-matc hing. If F > G , then F and G satisfy condition (ii), a con tradiction. Hence F < G , and therefore | G | = 2. Moreo v er, F / ∈ U , since otherwise ( U \ { F } ) ∪ { G } > lex U , con tradicting the maximalit y of U . Since replacing F by G yields a ν -matc hing, the assumption forces F = G . Let M = { F 1 , . . . , F ν } and N = { G 1 , . . . , G ν } . By Remark 3.7 there exist i, j ∈ [ ν ] such that F i ∩ G j  = ∅ , F i  = G j , and F i ∩ G k = ∅ for all k  = j . After relabeling we ma y assume i = j = 1. If F 1 ∈ U , then W = ( N \ G 1 ) ∪ F 1 satisﬁes the conclusion, so assume F 1 / ∈ U . W e may further assume that no facet of M ∩ U can b e used directly . Namely , for every F ∈ M ∩ U , either F ∈ N or   { G ∈ N : F ∩ G  = ∅}   ≥ 2 . (9) Indeed, if F ∈ M ∩ U in tersects exactly one facet G ∈ N , then W = ( N \ G ) ∪ F satisﬁes the conclusion. W e argue by decreasing induction on n = | M ∩ U | . Step 1: L et n = ν − 1 . Then all facets of M except F 1 lie in U . W e ﬁrst sho w G 1 / ∈ U . If G 1 ∈ U , then N ′ = ( N \ G 1 ) ∪ F 1 is a ν -matching and M  = N ′ b ecause M > ℓ N . Applying Remark 3.7 to M and N ′ yields F i with i  = 1 intersecting exactly one facet of N ′ . Since F i , G 1 ∈ U , we hav e F i ∩ G 1 = ∅ . Hence F i in tersects exactly one facet of N , con tradicting the assumption (9). Hence G 1 / ∈ U . By Lemma 4.11 there exists H ∈ U such that H ∩ F 1  = ∅ and H ∩ G 1  = ∅ . Since F 1 ∈ M \ U , w e ha v e H / ∈ M , and hence U = ( M \ F 1 ) ∪ H . If H intersected at least tw o facets of N , then by assumption (9), the bipartite graph B ( U,N ) (Construction 3.1) w ould not be a tree, contradicting Lemma 3.3. Hence H intersects exactly one facet of N , namely G 1 . Thus W = ( N \ G 1 ) ∪ H is a ν -matching. It remains to show that H \ G 1 con tains a v ertex in V M \ V N . If | F 1 ∩ G 1 | = 1, say F 1 ∩ G 1 = { s } . Since | F 1 | , | G 1 | ≥ 3 due to (8), Remark 4.8(2) implies deg T ( s ) = 2, so H = N T [ s ]. Hence H \ G 1 is a singleton and H \ G 1 ⊆ F 1 \ G 1 . Because F 1 in tersects no facet of N other than G 1 , we ha ve F 1 \ G 1 ⊆ V M \ V N , which pro ves the claim. It remains to sho w that | F 1 ∩ G 1 | = 2 cannot o ccur. Supp ose | F 1 ∩ G 1 | = 2. Then there exists a path i 1 , i 2 , i 3 , i 4 in T such that F 1 = N T [ i 2 ] and G 1 = N T [ i 3 ], and either H = N T [ i 1 ] or H = N T [ i 4 ]. First assume H = N T [ i 1 ]. Since n = ν − 1, F do es not in tersect any facet of U other than H , so we hav e i 3 / ∈ V U . Because N T [ i 1 ] ∩ N T [ i 3 ] = { i 2 } and H intersects no facet of N other than G 1 = N T [ i 3 ], Remark 4.8(2) applied to N and U yields deg T ( i 2 ) = 2. Since i 3 / ∈ V U , the maximalit y of U implies N T [ i 1 ] > N T [ i 2 ], otherwise ( U \ N T [ i 1 ]) ∪ N T [ i 2 ] > lex U . Consequently | N T [ i 1 ] | = 3, and level of i 1 is greater than level of i 2 in T . Therefore N T [ i 2 ] > N T [ i 3 ], con tradicting (ii) and hence H  = N T [ i 1 ]. Then H = N T [ i 4 ]. By a symmetric argument we obtain N T [ i 2 ] < N T [ i 3 ]. Since M \ N T [ i 2 ]  = N \ N T [ i 3 ] (otherwise N > ℓ M ), applying Remark 3.7 to M ′ = ( M \ N T [ i 2 ]) ∪ N T [ i 3 ] and N con tradicts (9). Assume that the assertion holds for all n ≥ k + 1, where k ≤ ν − 2, and consider the case n = k . W rite M = { F 1 , . . . , F ν − k , F ′ 1 , . . . , F ′ k } , where F ′ p ∈ U for p = 1 , . . . , k and F q / ∈ U for q = 1 , . . . , ν − k . Recall that, by Remark 3.7, F 1 in tersects exactly one facet of N , namely G 1 . Hence ( N \ G 1 ) ∪ F 1 is a ν -matching of N ( T ). 19 Applying Remark 3.7 to M and the resulting matching, and rep eating this pro cedure whenever p ossible, we obtain a ν -matching N ∗ suc h that (a)   { F 1 , . . . , F ν − k } ∩ N ∗   = ν − k − 1; (b) for ev ery F ′ ∈ M ∩ U , either F ′ ∈ N ∗ or   { G ∈ N ∗ : F ′ ∩ G  = ∅}   ≥ 2. Applying Remark 3.7 once more to M and N ∗ , and relab eling if necessary , w e may assume that F ν − k ∩ G 2  = ∅ and F ν − k do es not in tersect any other facet of N ∗ . If G 2 ∈ U , then G 2 do es not intersect any facet of M other than F ν − k . Hence M ′ = ( M \ F ν − k ) ∪ G 2 is a ν -matc hing of N ( T ) with | M ′ ∩ U | = k + 1. Since M ′ and N satisfy conditions (i) and (ii), the induction h yp othesis yields the desired conclusion b ecause V M ′ \ V N ⊆ V M \ V N . No w, supp ose that G 2 / ∈ U . By Lemma 4.11 there exists H ′ ∈ U such that H ′ ∩ F ν − k  = ∅ and H ′ ∩ G 2  = ∅ . W e distinguish the following t w o cases. Case 1. Supp ose | F ν − k ∩ G 2 | = 1. Set F ν − k = N T [ i 1 ], G 2 = N T [ i 3 ], and F ν − k ∩ G 2 = { i 2 } , where i 1 , i 2 , i 3 form a path in T . Since | F ν − k | , | G 2 | ≥ 3 by (8), applying Remark 4.8(2) to M and N ∗ yields deg T ( i 2 ) = 2. Hence H ′ = N T [ i 2 ] = { i 1 , i 2 , i 3 } ∈ U , and it do es not intersect an y facet in M ∩ U . Moreov er, as i 1 , i 2 ∈ F ν − k and i 3 ∈ G 2 , the facet N T [ i 2 ] do es not in tersect an y facet in N ∗ ∩ U . Thus M ′′ = ( M \ F ν − k ) ∪ H ′ is a ν -matching with | M ′′ ∩ U | = k + 1, V M ′′ \ V N ⊆ V M \ V N . Moreov er, M ′′ and N satisfy conditions (i) and (ii). Th us, the conclusion follo ws from the induction h yp othesis on n . Case 2. Supp ose | F ν − k ∩ G 2 | = 2. Set F ν − k = N T [ i 2 ], G 2 = N T [ i 3 ], and choose i 1 ∈ F ν − k \ G 2 and i 4 ∈ G 2 \ F ν − k , where i 1 , i 2 , i 3 , i 4 form a path in T . Then either H ′ = N T [ i 1 ] or H ′ = N T [ i 4 ]. Since T satisﬁes (C2), either | N T [ i 2 ] | = 3 or | N T [ i 3 ] | = 3. W e ﬁrst note that N T [ i 2 ] < N T [ i 3 ]. Otherwise | N T [ i 2 ] | = 3 and N T [ i 2 ] \ N T [ i 3 ] = { i 1 } with i 1 / ∈ V N , con tradicting (ii). Hence | N T [ i 3 ] | = 3. Supp ose that N T [ i 4 ] / ∈ U . Then H ′ = N T [ i 1 ]. Let T 1 and T 2 b e the connected comp onents of T \ i 3 with i 2 ∈ V ( T 1 ) and i 4 ∈ V ( T 2 ). Since N T [ i 2 ] , N T [ i 3 ] , N T [ i 4 ] / ∈ U , w e ha ve i 3 / ∈ V U . W e claim that | U T 1 | = | N T 1 | + 1. Let S 1 , . . . , S r b e the connected comp onen ts of T \ i 2 , with i 3 ∈ S 1 . Since i 3 / ∈ V U , Lemma 3.9(iii) giv es | U S 1 | = | M S 1 | . Moreo v er, i 2 ∈ V N ∗ and (N T [ i 2 ] \ N T [ i 3 ]) ∩ V N ∗ = ∅ , so Lemma 3.9(ii) implies | N S 1 | = | M S 1 | . Hence | U S 1 | = | N S 1 | , which yields | U T 2 | = | N T 2 | . Applying Lemma 3.9(iii) to the decomp osition T \ i 3 no w gives | U T 1 | = | N T 1 | + 1. By Remark 4.8(1) this giv es deg T ( i 2 ) = 2, and hence ( U \ N T [ i 1 ]) ∪ N T [ i 2 ] > U, con tradicting the maximalit y of U . Th us, this case cannot o ccur. Therefore N T [ i 4 ] ∈ U . Since i 4 ∈ G 2 ∈ N ∗ , it do es not lie in F 1 , . . . , F ν − k − 1 , and b ecause N T [ i 4 ] ∩ F ν − k  = ∅ , we hav e N T [ i 4 ] / ∈ M ∩ U . Hence i 4 do es not b elong to an y facet of M ∩ U . Th us, i 4 / ∈ V M , and hence ¯ M = ( M \ F ν − k ) ∪ G 2 is a ν -matc hing with F ν − k > G 2 and F ν − k / ∈ U , con tradicting our assumption on M . This shows that this case cannot o ccur. □ W e are now ready to pro ve the suﬃcient condition for the ν -th squarefree p ow er of the closed neigh b orho o d ideal of a tree to b e comp onent wise linear. Theorem 4.13. L et T b e a tr e e satisfying (C1) and (C2) , and set J = N I ( T ) . Then J [ ν ] has line ar quotients and is c omp onentwise line ar. Pr o of. Let U b e the unique ν -matc hing maximal with resp ect to > lex . Using Notation 4.6, w e order the generators of G ( J [ ν ] ) as u 1 > ℓ · · · > ℓ u n . W e prov e that J [ ν ] has linear quotients with resp ect to this order. Let M and N b e the ν -matc hings corresp onding to u a and u b , respectively , 20 with M > ℓ N . By [10, Corollary 8.2.4], it suﬃces to sho w that there exists a ν -matching W with W > ℓ N suc h that V W \ V N is a singleton and V W \ V N ⊆ V M \ V N . Since M > ℓ N , we hav e β ( M ) ≥ β ( N ). If β ( M ) > β ( N ), then there exist facets F ∈ M and G ∈ N with | F | = 2 and | G | ≥ 3 suc h that F intersects only G among the facets of N . Hence W = ( N \ G ) ∪ F is a ν -matching. Since | F | = 2, we ha v e β ( W ) > β ( N ) and thus W > ℓ N . Moreo v er V W \ V N is a singleton and V W \ V N ⊆ V M \ V N , as required. W e may therefore assume β ( M ) = β ( N ). If there exist facets F ∈ M and G ∈ N with F > G suc h that F \ G is a singleton and F \ G / ∈ V N , that is, F intersects no facet of N other than G . Th us W = ( N \ G ) ∪ F is a ν -matc hing, and since F > G we obtain W > ℓ N . As F \ G is a singleton, the required condition holds. Hence w e may assume that no suc h pair of facets exists. By Prop osition 4.12, there exist facets H ∈ U and G ∈ N , and a v ertex x ∈ H \ G , suc h that W = ( N \ G ) ∪ H is a ν -matching and x ∈ V M \ V N . In particular, level U ( W ) = level U ( N ) − 1, and therefore W > ℓ N . There are t w o p ossibilities for H : either H ∈ M or H / ∈ M . Case A. Supp ose that H / ∈ M . As sho wn in the pro of of Prop osition 4.12, the set H \ G is a singleton and W satisﬁes the required conditions. Case B. Suppose that H ∈ M . W e distinguish t w o sub cases according to | H ∩ G | . Case B.I. Assume | H ∩ G | = 2. By Proposition 4.9, | H | ≤ 3, and | H | = 2 is impossible since H and G are distinct facets of N ( T ). Hence | H | = 3. Thus | H \ G | = 1, so V W \ V N is a singleton. Since H ∈ M , we hav e V W \ V N ⊆ V M \ V N . Case B.II. Assume that | H ∩ G | = 1. By Prop osition 4.9, w e hav e | H | ≤ 3. If | H | = 2, then H \ G is a singleton and H \ G / ∈ V N , con tradicting our assumption. Hence | H | = 3. If | G | = 2, then ( U \ H ) ∪ G > lex U , contradicting the maximality of U . Thus | G | ≥ 3. Set H = N T [ i 1 ] and G = N T [ i 3 ] with H ∩ G = { i 2 } , where i 1 , i 2 , i 3 form a path in T . Since W and N diﬀer in one facet, Condition (C1) implies deg T ( i 2 ) = 2. Consider W ′ = ( N \ N T [ i 3 ]) ∪ N T [ i 2 ] . Since H \ G / ∈ V N , we obtain N T [ i 1 ] ∩ V N \ N T [ i 3 ] = ∅ . Also, N T [ i 2 ] ⊂ V { N T [ i 1 ] , N T [ i 3 ] } giv es N T [ i 2 ] ∩ V N \ N T [ i 3 ] = ∅ . Thus W ′ is a ν -matc hing of N ( T ), and clearly level( W ′ ) = lev el( N ). W e claim that W ′ > lex N . Once this is established, then W ′ > ℓ N , and the fact that | N T [ i 2 ] \ N T [ i 3 ] | = 1 implies that V W ′ \ V N is a singleton. Since N T [ i 2 ] \ N T [ i 3 ] ⊆ N T [ i 1 ] \ N T [ i 3 ] ∈ V M , w e obtain V W ′ \ V N ⊆ V W \ V N ⊆ V M \ V N , as required. No w w e pro ve the claim. W rite N T [ i 2 ] = F i,j,k and N T [ i 3 ] = F i ′ ,j ′ ,k ′ (see Notation 4.3). Since i = | N T [ i 2 ] | = 3, we hav e i ≤ i ′ . If i < i ′ , then N T [ i 2 ] > N T [ i 3 ], and hence W ′ > lex N , as required. No w supp ose i = i ′ = 3. By Prop osition 4.10, the lev el of i 1 in T is strictly less than that of i 3 . Since i 2 lies on the path b etw een i 1 and i 3 , we obtain j < j ′ , and hence W ′ > lex N , as required. □ W e conclude this section by presenting a class of graphs whose highest non-v anishing squarefree p o wer of the closed neighborho o d ideal is comp onent wise linear. Corollary 4.14. If G is a p ath gr aph or a whisker e d p ath gr aph (that is, a gr aph obtaine d by attaching a whicker to every vertex of a p ath gr aph), then the ide al N I ( G ) [ ν ] is c omp onentwise line ar. 21 5. Regularity of the highest squarefree power In this section, we demonstrate that if either Condition (C1) or (C2) is not satisﬁed, then one can construct a tree G for which reg ( N I ( G ) [ ν ] ) is arbitrarily larger than deg( N I ( G ) [ ν ] ). This pro vides a p ositive answer to Question 1.2. W e b egin b y deﬁning a class of trees that do es not satisfy Condition (C2). Example 5.1. Let G 1 b e the graph deﬁned in Figure 5 with n ≥ 1. It is easy to v erify that the matc hing num ber of N ( G 1 ) is 2( n + 1) + 1. Moreo ver, M 1 = n +1 [ i =1 { N G 1 [ r ′′ i ] , N G 1 [ s ′′ i ] } ∪ { N G 1 [ r ] } and M 2 = n +1 [ i =1 { N G 1 [ r ′′ i ] , N G 1 [ s ′′ i ] } ∪ { N G 1 [ s ] } are the only 2 n + 3-matchings of N ( G 1 ). F urthermore, G 1 satisﬁes Condition (C1), but do es not satisfy Condition (C2). r s r 1 r 2 r 3 r n +1 s 1 s 2 s 3 s n +1 r ′ 1 r ′′ 1 r ′ 2 r ′′ 2 r ′ 3 r ′′ 3 r ′ n +1 r ′′ n +1 s ′ 1 s ′′ 1 s ′ 2 s ′′ 2 s ′ 3 s ′′ 3 s ′ n +1 s ′′ n +1 · · · · · · G 1 Figure 5. The graph G 1 . Theorem 5.2. L et m b e a p ositive inte ger. Then ther e exists a tr e e G that satisﬁes Condition (C1) but fails to satisfy Condition (C2) , such that reg( N I ( G ) [ ν ] ) − deg( N I ( G ) [ ν ] ) = m . Pr o of. Fix a p ositive integer m . Let G b e a graph deﬁned in Example 5.1 with n = m . Then it follo ws that N I ( G ) [ ν ] = ( x V M 1 , x V M 2 ), where M 1 and M 2 are deﬁned in Example 5.1. It is clear that deg ( N I ( G ) [ ν ] ) = 5( m + 1) + 2. Consider the follo wing short exact sequence 0 − → S (( x V M 1 ) : x V M 2 ) ( − (5( m + 1) + 2)) − → S ( x V M 1 ) − → S N I ( G ) [ ν ] − → 0 . (10) Observ e that (( x V M 1 ) : x V M 2 ) = ( x A ), where A = { r 1 , . . . , r m +1 } . It is easy to see that reg( S/ ( x V M 1 )) = 5( m + 1) + 1 and reg( S / (( x V M 2 ) : x V M 1 )) = m . Then, applying Lemma 2.5 to Equation (10) yields reg( S /N I ( G ) [ ν ] ) = 5( m + 1) + m + 1. Hence, reg( N I ( G ) [ ν ] ) − deg( N I ( G ) [ ν ] ) = m , as desired. □ In the follo wing, we deﬁne the class of trees that do es not satisfy Condition (C1). 22 Example 5.3. Let G 2 b e the graph deﬁned in Figure 6 with n ≥ 1. It is easy to see that the matc hing num ber of N ( G 2 ) is 4 n + 1. Moreo ver, M 1 = n [ i =1 { N G 2 [ r ′′ i ] , N G 2 [ ˜ ˜ r i ] , N G 2 [ s ′′ i ] , N G 2 [ ˜ ˜ s i ] } ∪ { N G 2 [ t 1 ] } , M 2 = n [ i =1 { N G 2 [ r ′′ i ] , N G 2 [ ˜ ˜ r i ] , N G 2 [ s ′′ i ] , N G 2 [ ˜ ˜ s i ] } ∪ { N G 2 [ r ] } , and M 3 = n [ i =1 { N G 2 [ r ′′ i ] , N G 2 [ ˜ ˜ r i ] , N G 2 [ s ′′ i ] , N G 2 [ ˜ ˜ s i ] } ∪ { N G 2 [ s ] } , are the 4 n + 1-matchings of N ( G 2 ). F urthermore, G 2 satisﬁes Condition (C2), whereas it fails to satisfy Condition (C1). r s r 1 r n s 1 s 2 s n r ′ 1 r ′′ 1 r 2 r ′ 2 r ′′ 2 ˜ r 2 ˜ ˜ r 2 r ′ n r ′′ n ˜ r n ˜ ˜ r n s ′ 2 s ′′ 2 ˜ s 2 ˜ ˜ s 2 s ′ n s ′′ n ˜ s n ˜ ˜ s n · · · · · · t t 1 ˜ r 1 ˜ ˜ r 1 s ′ 1 s ′′ 1 ˜ s 1 ˜ ˜ s 1 G 2 Figure 6. The graph G 2 Theorem 5.4. L et m b e a p ositive inte ger. Then ther e exists a tr e e G that satisﬁes Condition (C2) but fails to satisfy Condition (C1) , such that reg( N I ( G ) [ ν ] ) − deg( N I ( G ) [ ν ] ) = m . Pr o of. Let m b e a p ositiv e integer. Let G b e a graph deﬁned in Example 5.3 with n = m . Then it follows that N I ( G ) [ ν ] = ( x V M 1 , x V M 2 , x V M 3 ), where M 1 , M 2 , and M 3 are deﬁned in Example 5.3. Since (( x V M 1 ) : x V M 2 ) = ( x t 1 ), it follo ws that the ideal ( x V M 1 , x V M 2 ) has linear quotients. Therefore, reg ( S/ ( x V M 1 , x V M 2 )) = 9 m + 1. Consider the follo wing short exact sequence 0 − → S (( x V M 1 , x V M 2 ) : x V M 3 ) ( − (9 m + 2)) − → S ( x V M 1 , x V M 2 ) − → S N I ( G ) [ ν ] − → 0 . (11) Observ e that (( x V M 1 , x V M 2 ) : x V M 3 ) = ( x t 1 , x A ), where A = { r , r 1 , . . . , r m } . Then, it follo ws that reg( S/ ( x V M 1 , x V M 2 ) : x V M 3 )) = m . Then, applying Lemma 2.5 to Equation (11) yields reg( S/N I ( G ) [ ν ] ) = 10 m + 1. Since deg( N I ( G ) [ ν ] ) = 9 m + 2, we conclude that reg ( N I ( G ) [ ν ] ) − deg( N I ( G ) [ ν ] ) = m . This completes the pro of. □ Next, w e study the regularit y of the ν -th square-free p ow er of the closed neigh b orho o d ideal of caterpillar graphs. W e b egin b y recalling the deﬁnition of a caterpillar graph. A graph G is said to b e a caterpillar graph if there exists a path P ⊆ G suc h that every vertex of G either lies on P or is adjacen t to a v ertex on P . The path P ⊆ G is referred to as the central path of G . 23 Theorem 5.5. L et G b e a c aterpil lar gr aph. Then, one has reg  S N I ( G ) [ ν ]  = deg( N I ( G ) [ ν ] ) − 1 . T o complete the pro of of Theorem 5.5, w e need to pro ve a couple of lemmas. W e begin b y ﬁxing some notation. Let I b e an ideal of S and let G ( I ) = { u 1 , . . . , u m } b e the minimal generating set. W e set I j = ( u 1 , . . . , u j ) for j = 1 , 2 , . . . , m . Lemma 5.6. L et G b e a c aterpil lar gr aph. L et I b e the ν -th squar efr e e p ower of the close d neighb orho o d ide al of G with G ( I ) = { u 1 , . . . , u m } . Then ther e exists a total or der on G ( I ) such that for al l j = 2 , . . . , m , the c olon ide al ( I j − 1 : u j ) is gener ate d either by variables or by variables to gether with distinct quadr atic monomials. Pr o of. F ollo wing Notation 4.3, we order the generators of I by u 1 > lex · · · > lex u m . F or each i , let M i denote the ν -matc hing asso ciated with u i . Let M a = { F i 1 ,j 1 ,k 1 , . . . , F i ν ,j ν ,k ν } and M b = { F i ′ 1 ,j ′ 1 ,k ′ 1 , . . . , F i ′ ν ,j ′ ν ,k ′ ν } b e tw o ν -matc hings of N ( G ) written in decreasing order with resp ect to > . Assume that a < b and set s = min { ℓ : F i ℓ ,j ℓ ,k ℓ  = F i ′ ℓ ,j ′ ℓ ,k ′ ℓ } . Since M a > lex M b , we ha v e F i s ,j s ,k s > F i ′ s ,j ′ s ,k ′ s . W e claim that F i s ,j s ,k s in tersects exactly one facet of M b . First observe that there exists t ≥ s suc h that F i s ,j s ,k s ∩ F i ′ t ,j ′ t ,k ′ t  = ∅ , since otherwise M b ∪ { F i s ,j s ,k s } would form a ( ν + 1)-matc hing of N ( G ), con tradicting the matching n um b er. Note that ev ery facet of N ( G ) has cardinality at most 3. If | F i s ,j s ,k s | = 3, then for all q ≥ s w e hav e F i s ,j s ,k s > F i ′ q ,j ′ q ,k ′ q , (12) and hence | F i ′ q ,j ′ q ,k ′ q | = 3 for all q ≥ s . If F i s ,j s ,k s in tersected tw o distinct facets of M b , say F i ′ p ,j ′ p ,k ′ p and F i ′ q ,j ′ q ,k ′ q with p, q ≥ s . Since these facets are contained in the cen tral path P , it follows that either F i ′ p ,j ′ p ,k ′ p > F i s ,j s ,k s or F i ′ q ,j ′ q ,k ′ q > F i s ,j s ,k s , con tradicting (12). Th us F i s ,j s ,k s in tersects exactly one facet of M b . If | F i s ,j s ,k s | = 2, the conclusion clearly holds. Let t be the unique index suc h that F i s ,j s ,k s ∩ F i ′ t ,j ′ t ,k ′ t  = ∅ . Then M c = ( { F i s ,j s ,k s } ∪ M b ) \ { F i ′ t ,j ′ t ,k ′ t } is a ν -matc hing of N ( G ), and since F i s ,j s ,k s > F i ′ t ,j ′ t ,k ′ t w e obtain M c > lex M b . Hence, for ev ery a < b there exists c < b such that M c and M b diﬀer b y exactly one facet. Since ev ery facet of N ( G ) has size at most 3, we hav e | V M c \ V M b | ≤ 2. This implies that the colon ideal ( I j − 1 : u j ) is generated either b y v ariables or b y v ariables together with quadratic monomials. It remains to show that the quadratic generators are distinct. Supp ose that x i 1 x i 2 and x i 3 x i 4 b elong to G ( I j − 1 : u j ) with |{ i 1 , i 2 } ∩ { i 3 , i 4 }| = 1. Then { i 1 , i 2 } and { i 3 , i 4 } are edges of G . Without loss of generalit y assume i 2 = i 3 , so that P ′ = {{ i 1 , i 2 } , { i 2 , i 4 }} forms a path in G , and ev ery vertex of P ′ lies outside V M j . If deg G ( i 2 ) = 2, then M j ∪ N G [ i 2 ] is a matc hing of N ( G ), contradicting the matching n um b er. If deg G ( i 2 ) ≥ 3, there exists i 5 ∈ N G [ i 2 ] \ { i 1 , i 4 } and again M j ∪ N G [ i 5 ] forms a matching, yielding the same con tradiction. Therefore { i 1 , i 2 } ∩ { i 3 , i 4 } = ∅ , which pro ves that the quadratic generators are distinct. □ Lemma 5.7. Under the assumption of L emma 5.6, let j ∈ [ m ] . If the c olon ide al ( I j − 1 : u j ) is gener ate d by variables to gether with k distinct quadr atic monomials, then ther e exists i > j such that deg ( u j ) + k − 1 < deg ( u i ) . 24 Pr o of. Let u 1 > lex · · · > lex u m b e the order on G ( I ) in tro duced in the pro of of Lemma 5.6, and let M i = { F i 1 , . . . , F i ν } denote the ν -matching asso ciated with u i . Assume that there exists j ∈ [ m ] suc h that the minimal generating set of the colon ideal ( I j − 1 : u j ) con tains exactly t w o distinct quadratic monomials x i 1 x i 2 and x i 3 x i 4 with the lev el of i 1 strictly les s than that of i 3 . Since x i 1 x i 2 ∈ G ( I j − 1 : u j ), it follows from the proof of Lemma 5.6 that there exists r < j such that M r and M j diﬀer by exactly one facet and ( u r : u j ) = x i 1 x i 2 . Hence there exist indices p and q such that M r \ { F r p } = M j \ { F j q } and F r p \ F j q = { i 1 , i 2 } . Moreo v er, b y minimality of the colon generator there is no facet F ∈ F ( N ( G )) suc h that F \ F j q ⊂ F r p \ F j q . Consequently , along the central path P = { . . . , i θ − 2 , i θ − 1 , i θ , i θ +1 , i θ +2 , . . . } of the caterpillar G w e hav e F r p = N G [ i θ − 1 ] and F j q = N G [ i θ +1 ], with de g G ( i θ ) ≥ 3. Note that deg G ( i θ − 2 ) = 2 and deg G ( i θ − 3 ) ≥ 2, otherwise, the facet corresp onding to a pendant vertex is disjoin t from V M j , whic h con tradicts the matching num ber of N ( G ). There exists a facet F j l ′ ∈ M j suc h that N G [ i θ − 2 ] ∩ F j l ′  = ∅ ; otherwise, M j ∪ { N G [ i θ − 2 ] } w ould form a matching of N ( G ), contradicting the matching num b er. Set N 1 = ( M j \ { F j l ′ } ) ∪ N G [ i θ − 2 ] . If | F j l ′ | = 2, w e set M i ′ = N 1 . Otherwise | F j l ′ | = 3, in whic h case F j l ′ = N G [ i θ − 4 ]. Rep eating the ab o ve argumen t along the path, w e construct successiv ely N t = ( N t − 1 \ { F j l } ) ∪ N G [ i θ − (3 t − 1) ] , where F j l denotes the corresp onding facet in M j . This pro cess terminates at some step t when the replaced facet has cardinalit y 2. Indeed, if all replaced facets had cardinality 3 and i θ − 3( t +1) w ere a p endan t v ertex of G , then N t ∪ { N G [ i θ − 3( t +1) ] } would form a matc hing of N ( G ), again con tradicting the matching n um b er. Let M i ′ = N t . By construction, |{ F ∈ M j : | F | = 2 }| > |{ F ∈ M i ′ : | F | = 2 }| , and hence M j > lex M i ′ . Moreo v er, | V M i ′ | = | V M j | + 1. Let A = { N G [ i θ − (3 k − 1) ] : k = 1 , . . . , t } . Since V A \ { i 1 , i 2 } ⊂ V M j while { i 3 , i 4 } ⊈ V M j , the v ertices i 3 and i 4 do not b elong to V A . Consequen tly , the colon ideal ( I i ′ − 1 : u i ′ ) contains the quadratic monomial x i 3 x i 4 , since the level of i 1 is less than that of i 3 . Rep eating the same argumen t, we obtain a matching M i ′′ suc h that M i ′ > lex M i ′′ and | V M i ′′ | = | V M i ′ | + 1 . Therefore, deg ( u i ′′ ) = deg( u j ) + 2, which completes the pro of. □ Pr o of of The or em 5.5. Let I b e the ν -th squarefree p ow er of the closed neigh b orho o d ideal of G and let G ( I ) = { u 1 , . . . , u m } . It is clear that reg( S/I ) ≥ max 1 ≤ i ≤ n { deg( u i ) } − 1. Th us it suﬃces to prov e that reg( S/I ) ≤ max 1 ≤ i ≤ n { deg( u i ) } − 1. Using Notation 4.3, w e order the elemen ts of G ( I ) by u 1 > lex · · · > lex u m . Set d j = deg ( u j ) for j = 1 , . . . , m . F or j = 2 , . . . , m consider the follo wing sequence of short exact sequences 0 − → S ( I j − 1 : u j ) ( − d j ) − → S I j − 1 − → S I j − → 0 . (13) 25 Applying Lemma 2.5 to Equation (13), yields reg( S/ ( I )) ≤ max { α , reg( S/I 1 ) } , where α = max 2 ≤ j ≤ m { reg( S/ ( I j − 1 : u j )) + d j − 1 } . By [19, Theorem 2.18], if ∆ is a 1-dimensional for- est, then reg( S/I (∆)) equals the maximal num ber of pairwise disjoint facets in ∆. Hence, by Lemma 5.6, the ideal ( I j − 1 : u j ) is generated b y v ariables and β j distinct quadratic monomials, where β j denotes the n um b er of quadratic generators. Consequently , reg ( S/ ( I j − 1 : u j )) = β j . Th us α = max 2 ≤ j ≤ m { β j + d j − 1 } . By Lemma 5.7, for every j there exists i > j such that β j + d j ≤ d i . Therefore, α ≤ max 2 ≤ j ≤ m { d j − 1 } . Since reg( S/I 1 ) = d 1 − 1, we conclude that reg( S/I ) ≤ max { max 2 ≤ j ≤ m ( d j − 1) , d 1 − 1 } = max 1 ≤ j ≤ m { d j − 1 } , as desired . □ References [1] C. Berge, Hyp ergraphs: Combinatorics of Finite Sets, Mathematical Library 45, North-Holland, 1989. [2] M. Bigdeli, J. Herzog, R. Zaare-Nahandi, On the index of p ow ers of edge ideals, Communications in Algebra, 46 (3), 1080-1095, 2018. [3] S. Chakrab orty , A. P . Joseph, A. Roy , A. Singh. 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Zheng, Standard graded vertex cov er algebras, cycles and leav es, T rans- actions of the American Mathematical So ciety , 360 (12), 6231-6249, 2008. [10] J. Herzog, T. Hibi, Monomial ideals, volume 260 of Graduate T exts in Mathematics, Springer-V erlag London, Ltd., London, 2011. [11] L. T. Hoa, N. D. T am, On some inv ariants of a mixed pro duct of ideals, Archiv der Mathematik, 94 , 327–337, 2010. [12] J. Honeycutt, S. K. Sather-W agstaﬀ, Closed neigh b orho o d ideals of ﬁnite simple graphs, La Matematica 1 , 387394, 2022. [13] E. Kam b eri, F. Nav arra, A. A. Qureshi, On squarefree p ow ers of facet ideals of simplicial trees, Journal of Algebra 693, pages 240-276, 2026. [14] M. Nasernejad, A. A. Qureshi, S. Bandari, A. Musapa¸ sao˘ glu, Dominating Ideals and Closed Neighborho o d Ideals of Graphs, Mediterranean Journal of Mathematics, 19 :Article no: 152, 2022. [15] M. Nasernejad, A. A. Qureshi, Algebraic implications of neighborho o d h yp ergraphs and their transv ersal h yp ergraphs, Communications in Algebra, 52 (6), 2328–2345, 2024. [16] M. Nasernejad, S. Bandari, L. G. Rob erts, Normality and asso ciated primes of closed neighborho o d ideals and dominating ideals, Journal of Algebra and its Applications, 24 (1), Paper No. 2550009, 19, 2025. [17] L. Sharifan, S. Moradi, Closed neighborho o d ideal of a graph, Ro cky Moun tain Journal of Mathematics 50 (3), 1097–1107, 2020. [18] L. Sharifan, t -closed neighborho o d ideal of a graph, Beitr¨ age zur Algebra und Geometrie, 66 , 927–940, 2025. [19] X. Zheng, Resolutions of F acet Ideals, Communications in Algebra, 32 (6), 2301–2324, 2004. Sabanci University, F acul ty of Engineering and Na tural Sciences, Or t a Mahalle, Tuzla, 34956, Ist anbul, Turkey Email address : amalore.p@gmail.com, amalore.pushparaj@sabanciuniv.edu Email address : aqureshi@sabanciuniv.edu, ayesha.asloob@sabanciuniv.edu

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