Disjoint Correspondence Colorings for $K_5$-Minor-free Graphs

Disjoin t Corresp ondence Colorings for K 5 -Minor-free Graphs W outer Cames v an Baten burg ∗ Daniel W. Cranston † F ran ti ˇ sek Kardo ˇ s ‡ F ebruary 19, 2026 Abstract Thomassen famously pro ved that every planar graph is 5-choosable. W e explore v ariants of this result, fo cusing on finding disjoint correspondence colorings, in the more general class of K 5 -minor-free graphs. Corresp ondence colorings generalize list colorings as follo ws. Given a graph G and a p ositiv e integer t , a corresp ondence t -cov er M assigns to each v ∈ V ( G ) a set of allo wable colors { 1 v , . . . , t v } and to eac h edge v w ∈ E ( G ) a matching b etw een { 1 v , . . . , t v } and { 1 w , . . . , t w } . An M -coloring φ pic ks for eac h vertex v a color φ ( v ) (from the set { 1 v , . . . , t v } ) suc h that for each edge v w ∈ E ( G ) the colors φ ( v ) , φ ( w ) are not matched to each other. Tw o M -colorings φ 1 , φ 2 of G are disjoint if φ 1 ( v )  = φ 2 ( v ) for all v ∈ V ( G ). F or ev ery K 5 -minor-free graph G and ev ery corresp ondence 6-co v er M of G , we construct 3 pairwise disjoin t M -colorings φ 1 , φ 2 , φ 3 . In contrast, we provide examples of K 5 -minor-free graphs and corresp ondence 5- co vers M that do not admit 3 disjoint M -colorings. 1 In tro duction 1.1 Definitions and Examples Corresp ondence coloring generalizes list coloring, and w as in tro duced to resolve a longstanding open question [12] on the list-chromatic n umber of certain planar graphs. Recall that a list assignment L for a graph G giv es to each v ertex v of G a set L ( v ) of allo wable colors. An L -c oloring is a prop er coloring φ suc h that φ ( v ) ∈ L ( v ) for all vertices v . If | L ( v ) | = t for all v , then L is called a t -assignment and an L -p acking consists of L -colorings φ 1 , . . . , φ t suc h that φ i ( v )  = φ j ( v ) for all v ertices v and all distinct i, j ∈ [ t ]. That is, the colors φ i ( v ) partition L ( v ) for each vertex v . F or a graph G , a c orr esp ondenc e t -c over M assigns to each v ∈ V ( G ) a set of allo wable colors { 1 v , . . . , t v } and to eac h edge v w ∈ E ( G ) a matching b et ween { 1 v , . . . , t v } and { 1 w , . . . , t w } . An M -coloring φ picks for eac h vertex v a color φ ( v ) (from the set { 1 v , . . . , t v } ) such that for eac h edge v w ∈ E ( G ) the colors φ ( v ) , φ ( w ) are not matched to eac h other. (It is straightforw ard to verify that corresp ondence coloring generalizes list coloring.) Two M -colorings φ 1 , φ 2 of G are disjoint if φ 1 ( v )  = φ 2 ( v ) for all v ∈ V ( G ). An M -packing for G consists of disjoin t M -colorings φ 1 , . . . , φ t . ∗ D ´ epartemen t d’Informatique, Universit ´ e libre de Bruxelles, Belgium; w.p.s.camesvanbatenburg@gmail.com . Supp orted by the Belgian National F und for Scientific Research (FNRS). † Departmen t of Mathematics, College of William & Mary , Williamsburg, V A, USA; dcransto@gmail.com . ‡ LaBRI, CNRS, Univ ersity of Bordeaux, T alence, F-33405, F rance; Departmen t of Computer Science, F ac- ult y of Mathematics, Physics and Informatics, Comenius Universit y , Mlynsk´ a Dolina, 842 48 Bratislav a, Slov akia; frantisek.kardos@u-bordeaux.fr . 1 The list-chr omatic numb er , χ ℓ ( G ), and list p acking numb er , χ ⋆ ℓ ( G ), are the minimum v alues of t suc h that for every t -assignment G admits, resp ectiv ely , a list coloring and a list packing. Analogously , for corresp ondence t -cov ers, we define the c orr esp ondenc e chr omatic numb er , χ c ( G ), and the c orr esp ondenc e p acking numb er , χ ⋆ c ( G ). List packing and corresp ondence packing w ere in tro duced in [4], where the authors discussed connections with other areas of discrete math. These parameters hav e since b een studied in numerous pap ers [1, 2, 3, 5, 6, 7, 10, 14, 15, 18]. As a warm up with these definitions, we provide a few examples. Note that alw ays χ ( G ) ⩽ χ ℓ ( G ) ⩽ χ c ( G ) ⩽ χ ⋆ c ( G ); and also χ ℓ ( G ) ⩽ χ ⋆ ℓ ( G ) ⩽ χ ⋆ c ( G ). F or a graph class G and a graph parameter f , let f ( G ) := max G ∈G f ( G ). Let F denote the class of forests, C the class of cycles, and C e the class of cycles of even length. So 2 = χ ( F ) = χ ℓ ( F ) = χ ⋆ ℓ ( F ) = χ c ( F ) = χ ⋆ c ( F ). The lo wer b ound is trivial, and the upp er b ound follo ws from Lemma A b elo w. It is well-kno wn (and easy to c heck) that χ ℓ ( C e ) = 2 < 3 = χ ℓ ( C ). But the reader ma y find the following more surpising: χ ⋆ ℓ ( C e ) = χ ⋆ ℓ ( C ) = 3, χ c ( C e ) = χ c ( C ) = 3, and χ ⋆ c ( C e ) = χ ⋆ c ( C ) = 4. In the first case, the upp er b ound requires a bit of work. But in the second and third cases, the upp er b ounds hold b ecause the graphs are 2-degenerate; in particular, the last of these follo ws from Lemma A b elo w. The lo wer b ounds on b oth χ c ( C e ) and χ ⋆ c ( C e ) come from constructing a cov er that violates a necessary parit y-type in v arian t needed to find the desired coloring or pac king. F or more details on these lo wer b ounds, see [3] or the introduction of [10]. 1.2 Main Result The authors of [4] proposed the problem of determining χ ⋆ c ( G ) for v arious graph classes G . Let K r denote the class of all graphs that are K r -minor-free. It is easy to chec k that χ ⋆ c ( K 2 ) = 1, χ ⋆ c ( K 3 ) = 2, and χ ⋆ c ( K 4 ) = 4; the final upp er b ound follo ws from Lemma A b elo w b ecause all graphs in K 4 are 2-degenerate. So the first op en case is χ ⋆ c ( K 5 ). The b est known upp er b ound is χ ⋆ c ( K 5 ) ⩽ 8. This follo ws from [5, Theorem 1.7] b ecause ev ery K 5 -minor-free graph has maxim um a verage degree less than 6. W e cautiously believe that this upper bound can be improv ed as follo ws. Conjecture 1. If G is K 5 -minor-fr e e and M is a c orr esp ondenc e 6-c over, then G has 6 disjoint M -c olorings. That is, χ ⋆ c ( K 5 ) = 6 . Dv o ˇ r´ ak, Norin, and P ostle [11] called a graph weighte d ϵ -flexible with respect to a list-assignmen t L if there exists a probability distribution on the L -colorings ϕ of G suc h that P ( ϕ ( v ) = c ) ⩾ ϵ for every vertex v and color c ∈ L ( v ). In other words, weigh ted ϵ -flexibility ensures that at eac h v ertex eac h color has a decent c hance to app ear in the random list-coloring. Dvo ˇ r´ ak, Norin, and P ostle pro ved that there is an absolute constant ϵ > 0 such that all planar graphs are w eighted ϵ -flexible with resp ect to every 7-assignment, and they p osed as an op en problem to show the same for 6-assignments or even 5-assignmen ts. While this problem remains wide op en, for 7-assignments the v alue of ϵ has b een improv ed considerably from 7 − 36 to 2 − 7 b y [14] and [2]. As detailed in [5] and [14], ev ery graph G is weigh ted 1 χ ⋆ ℓ ( G ) -flexible with resp ect to every χ ⋆ ℓ ( G )-assignmen t. Th us a pro of of Conjecture 1 will make significant progress: it will imply that every planar graph (which is K 5 -minor-free) is weigh ted 1 6 -flexible with resp ect to every 6-assignmen t. The main goal of this note is to prov e the follo wing result, in supp ort of the ab ov e conjecture. Main Theorem. L et G b e a K 5 -minor-fr e e gr aph. (i) F or every c orr esp ondenc e 6-c over M of G , ther e exist disjoint M -c olorings φ 1 , φ 2 , φ 3 . That is, for al l v ∈ V ( G ) the c olors φ 1 ( v ) , φ 2 ( v ) , φ 3 ( v ) ar e p airwise distinct. (ii) But the analo gous statement is false if we r eplac e 6-c over with 5-c over. 2 The fact that K 5 -minor-free graphs are 5-degenerate easily implies that they ha ve list-c hromatic n umber at most 6, which is close to the optimal b ound 5. How ever, for the corresp ondence packing n umber this approac h performs quite p o orly . As discussed ab o ve, χ ⋆ c ( C ) ⩽ 4 due to the 2-degeneracy of cycles. More generally , by Lemma A we ha ve χ ⋆ c ( G ) ⩽ 2 d for ev ery d -degenerate graph G , and this is sharp [4, Prop. 24] for ev ery d ⩾ 1, due to large complete bipartite graphs. Therefore a v anilla degeneracy argumen t can merely yield χ ⋆ c ( K 5 ) ⩽ 10, while it is kno wn that χ ⋆ c ( K 5 ) ⩽ 8. This indicates that establishing Conjecture 1 is nontrivial. On the low er b ound side, note that the second part of our Main Theorem implies that χ ⋆ c ( K 5 ) ⩾ 6. 1.3 Pro of ov erview When constructing disjoin t list colorings and corresp ondence colorings, it is often conv enient to pro ceed b y induction. Given a graph G , a corresp ondence t -co ver M of G and disjoin t partial M -colorings φ 0 1 , . . . , φ 0 s , for each v ertex v that is not y et colored we seek a color φ j ( v ) to extend φ 0 j to v . W e write L j ( v ) for the subset of [ t ] that is not yet forbidden from use as φ j ( v ), due to the colors already used on N ( v ) and the matchings M . The follo wing lemma, from [4], is generally useful for handling vertices of low degree. Its pro of is instructive, so w e include it. Lemma A ([4]) . L et G b e a gr aph with a c orr esp ondenc e t -c over M and fix v ∈ V ( G ) . If d ( v ) ⩽ t/ 2 , then any disjoint M -c olorings φ ′ 1 , . . . , φ ′ s of G − v extend to disjoint M -c olorings φ 1 , . . . , φ s of G . Pr o of. W e build an auxiliary bipartite graph B with 1 , . . . , t as the v ertices of the first part, and φ ′ 1 , . . . , φ ′ s as the vertices of the second part, and iφ ′ j ∈ E ( B ) if i ∈ L j ( v ). T o extend the M -colorings φ ′ j , we seek a matching M B saturating the second part. F or this we use Hall’s Theorem. Note that d B ( i ) ⩾ s − d ( v ) and d B ( φ ′ j ) ⩾ t − d ( v ), for all i ∈ [ t ] and j ∈ [ s ]. Consider S ⊆ { φ ′ 1 , . . . , φ ′ s } . Clearly | N B ( S ) | ⩾ t − d ( v ) ⩾ t/ 2. So if | S | ⩽ t/ 2, then | N B ( S ) | ⩾ | S | as desired. Otherwise, | S | > t/ 2 ⩾ d ( v ). So | S | + d B ( i ) ⩾ | S | + s − d ( v ) > s b ecause | S | > d ( v ). So b y Pigeonhole i ∈ N B ( S ) for all i ∈ [ t ]. That is, | N B ( S ) | = t ⩾ | S | , as desired. Th us, b y Hall’s Theorem we hav e the desired matc hing M B in B . By definition, eac h matching in M need not be p erfect. But to pro v e the results in this paper, it suffices to consider the case that all matchings are p erfect; if not, then we can grow them greedily . (In what follows, we implicitly assume that all matchings in M are p erfect.) An r -sum of graphs G 1 and G 2 is formed from the disjoin t union G 1 + G 2 b y sp ecifying an r -clique x 1 , . . . , x r in G 1 and an r -clique y 1 , . . . , y r in G 2 , iden tifying x i with y i for each i ∈ [ r ], and p ossibly deleting some edges of the resulting r -clique. The Wagner gr aph M 8 (also called the M¨ obius 8-ladder ) is formed from the 8-cycle z 1 · · · z 8 b y adding the 4 chords z i z i +4 for each i ∈ [4]. W agner [21] ch aracterized the class of maximal K 5 -minor-free graphs as follows. Theorem B ([21]) . If a gr aph G is maximal K 5 -minor-fr e e, then G c an b e built by a se quenc e of 2 -sums and 3 -sums, without deleting e dges, starting fr om maximal planar gr aphs and c opies of M 8 . So to prov e our Main Theorem, we would lik e to reduce to the case of planar graphs and copies of M 8 . How ever, w e m ust ensure that our disjoint M -colorings φ 1 , φ 2 , φ 3 agree on the copies of K 2 and K 3 where we p erform our 2-sums and 3-sums. This motiv ates the following sligh tly stronger lemmas. W e handle the copies of M 8 first, even for 6 disjoint M -colorings, since this step is easy . 3 Lemma 2. Given a c opy of M 8 and a c orr esp ondenc e 6-c over M of M 8 , ther e exist disjoint M - c olorings φ 1 , . . . , φ 6 . F urthermor e, if φ 0 1 , . . . , φ 0 6 ar e disjoint M -c olorings of a sp e cifie d K 2 in M 8 , then we c an cho ose φ 1 , . . . , φ 6 to extend φ 0 1 , . . . , φ 0 6 . Pr o of. W e use induction on the num b er n ′ of uncolored v ertices. The base case, n ′ = 0, is trivial. And the induction step holds, b ecause M 8 is 3-degenerate, by Lemma A with s := 6 and t := 6. No w we consider planar graphs. Lemma 3. L et G b e a planar gr aph with a 6-c orr esp ondenc e c over M . If C is a K 3 in G , then e ach choic e φ 0 1 , φ 0 2 , φ 0 3 of disjoint M -c olorings of C extends to disjoint M -c olorings φ 1 , φ 2 , φ 3 of G . The k ey to the pro of of Lemma 3 is proving a still stronger statement, given b elo w in the Key Lemma, that more easily facilitates pro of by induction. Our tec hnique is known as a “Thomassen- st yle” pro of, due to the strikingly short and elegan t proof of Carsten Thomassen [19] that all planar graphs are 5-choosable. This metho d has since b een applied many times [8, 13, 16, 17, 20, 22, 23]. F or a unified study of n umerous such examples, see [9, Chapter 11]. Key Lemma. L et G b e a plane ne ar-triangulation, with a c orr esp ondenc e 6-c over M . Denote the vertic es on the outer fac e of G , in clo ckwise or der, by w 1 , . . . , w n . Ther e exist disjoint M - c olorings φ 1 , φ 2 , φ 3 that extend φ 0 1 , φ 0 2 , φ 0 3 and that have φ j ( v ) ∈ L j ( v ) for al l j ∈ [3] if φ 0 1 , φ 0 2 , φ 0 3 and L 1 , L 2 , L 3 satisfy the fol lowing c onditions (1) L 1 ( v ) = L 2 ( v ) = L 3 ( v ) = [6] for e ach vertex v not on the outer fac e. (2) F or e ach vertex w i with i ∈ { 3 , . . . , n } ther e exist distinct c olors c 1 , . . . , c 6 ∈ [6] with L 1 ( w i ) = { c 1 , c 2 , c 3 , c 4 } and L 2 ( w i ) = { c 1 , c 2 , c 5 , c 6 } and L 3 ( w i ) = { c 3 , c 4 , c 5 , c 6 } . (3) φ 0 1 , φ 0 2 , φ 0 3 ar e disjoint M -c olorings of G [ { w 1 , w 2 } ] . 2 Pro ofs In this section we present pro ofs of our results. W e first pro ve part (i) of our Main Theorem, by using Lemma 2, Lemma 3, and Theorem B (W agner’s characterization of K 5 -minor-free graphs). That is, given a K 5 -minor-free graph G and a corresp ondence 6-co ver M , w e sho w how to construct disjoin t M -colorings φ 1 , φ 2 , φ 3 . Next, we prov e Lemma 3 using our Key Lemma; and after this we pro ve our Key Lemma via a “Thomassen-style” induction pro of. Finally , w e conclude this section b y proving part (ii) of our Main Theorem. That is, we construct a K 5 -minor-free graph G and a corresp ondence 6-cov er M of G that do not admit 3 disjoint M -colorings. Pr o of of the Main The or em. Fix a K 5 -minor-free graph G and a corresp ondence 6-cov er M . It suffices to consider the case that G is a maximal K 5 -minor-free graph; if not, then w e add edges to mak e it so. (The trivial case that | V ( G ) | ⩽ 2 is handled easily by Lemma A.) By Theorem B, G can b e constructed b y a sequence of 2-sums and 3-sums starting from copies of M 8 and planar graphs. F urthermore, b y possibly reordering the sequence, w e can assume that for eac h 2-sum or 3-sum one of the graphs being summed is itself either a cop y of M 8 or a planar graph. Fix such a construction sequence, and let q b e the sum of its num b ers of 2-sums and 3-sums. W e pro ceed by induction on q . If q = 0, then G is planar or a copy of M 8 , so we are done by Lemma 2 4 or by Lemma 3. T o b e precise, in the latter case, w e should first fix a cop y C of K 3 and disjoint M -colorings of C . But this is easy by Lemma A. This completes the base case, q = 0. No w assume that q ⩾ 1. So G is formed from a 2-sum or 3-sum of graphs G 1 and G 2 , where G 2 is planar or a copy of M 8 and G 1 is formed from a sequence of 2-sums and 3-sums of length q − 1. By the induction h yp othesis, G 1 has disjoint M -colorings φ ′ 1 , φ ′ 2 , φ ′ 3 . Let the cop y of K 2 or K 3 in G 2 , that is inv olved in the sum to form G , inherit from the corresp onding v ertices in G 1 three disjoin t M -colorings φ 0 1 , φ 0 2 , φ 0 3 . By Lemma 2 or Lemma 3, w e can extend φ 0 1 , φ 0 2 , φ 0 3 to disjoin t M -colorings φ ′′ 1 , φ ′′ 2 , φ ′′ 3 of G 2 . (If G 2 is planar and only a K 2 is precolored, then we first extend the precoloring to a K 3 , via Lemma A, b efore inv oking Lemma 3; this is p ossible b ecause G 2 is maximal planar , so every K 2 is contained in a K 3 .) Since φ ′ j and φ ′′ j agree on the vertices of V ( G 1 ) ∩ V ( G 2 ), for all j ∈ [3], together they give our disjoin t M -colorings φ 1 , φ 2 , φ 3 of G . Note that if we could prov e a stronger version of Lemma 3 with 4 disjoint M -colorings (or 5 or 6), then the ab ov e pro of w ould give an an analogously stronger version of the Main Theorem. Pr o of of Lemma 3. W e now use the Key Lemma to prov e Lemma 3. Fix a planar graph G with a corresp ondence 6-co ver M , a cop y C of K 3 in G , and disjoint M -colorings φ 0 1 , φ 0 2 , φ 0 3 of C . W e use induction on | V ( G ) | ; the base case | V ( G ) | = 3 holds trivially . First supp ose that C is a separating cycle in G . F or eac h comp onen t H i of G − V ( C ), let G i := G [ V ( H i ) ∪ V ( C )]. By the induction hypothesis, the lemma holds for each G i ; that is, eac h G i has disjoint M -colorings φ i 1 , φ i 2 , φ i 3 that agree with φ 0 1 , φ 0 2 , φ 0 3 on C . Since all these disjoin t M -colorings agree on C , their union gives the desired disjoin t M -colorings of G . So assume instead that C is not a separating cycle in G ; that is, C is the b oundary of a face. By redra wing if needed, we assume that C is the boundary of the outer face of G ; w e denote its v ertices b y w 1 , w 2 , w 3 . Let G ′ := G − w 3 . F or eac h x ∈ N G ( w 3 ) and eac h i ∈ [3], w e delete from L i ( x ) the color φ 0 i ( w 3 ) as well as another arbitrary color so that x will satisfy condition (2) of the Key Lemma. Now conditions (1), (2), and (3) of the Key Lemma hold. So we can extend these disjoin t partial M -colorings to disjoint M -colorings φ 1 , φ 2 , φ 3 of G that agree on C with φ 0 1 , φ 0 2 , φ 0 3 . No w we prov e our Key Lemma. T o denote the matching of M for an edge y z ∈ E ( G ), we write M y z . F urthermore, given a partial M -coloring φ i that colors a v ertex y , we denote b y M y z ( φ i ( y )) the color for z that is matched by M y z to the color φ i ( y ) for y . (And for an arbitrary subset S of [6] we let M y z ( S ) := S i ∈ S M y z ( i ).) Pr o of of the Key L emma. Our pro of is by induction on | V ( G ) | . The base case, | V ( G ) | = 3, is handled as follo ws. W e construct a bipartite graph B with the vertices of one part b eing φ 1 , φ 2 , φ 3 , the vertices of the other part b eing the colors 1 , . . . , 6 and φ j b eing adjacent to a color h if h ∈ L j ( w 3 ) \ { M w 1 w 3 ( φ 0 j ( w 1 )) , M w 2 w 3 ( φ 0 j ( w 2 )) } . That is, φ j is not adjacent to a color h precisely when h is either absent from L j or when the color is “blo c k ed” from b eing used on w 3 b y φ j , due to the color used on w 1 or on w 2 . It suffices to find a matching in B that saturates { φ 1 , φ 2 , φ 3 } ; this will tell us ho w to extend the disjoint M -colorings to w 3 . W e will do this b y Hall’s Theorem. Note that d B ( φ j ) ⩾ | L j ( v 3 ) | − 2 = 2 for all j ∈ [3]. F urthermore, L 1 ( w 3 ) ∩ L 2 ( w 3 ) ∩ L 3 ( w 3 ) = ∅ . Hence, | N B ( φ 1 ) ∪ N B ( φ 2 ) ∪ N B ( φ 3 ) | ⩾ 3. Thus, B satisfies the hypothesis of Hall’s Theorem, and we get the desired matching in B saturating { φ 1 , φ 2 , φ 3 } . No w we consider the induction step. First supp ose that G contains a c hord w i w ℓ of the cycle b ounding the outer face. By symmetry , we assume that 1 / ∈ { i, ℓ } (otherwise w e relab el vertices in coun terclo c kwise order, maintaining the fact that | L j ( w h ) | = 1 for eac h h ∈ [2] and each j ∈ [3]). 5 Let C 1 b e the cycle induced b y w 1 · · · w i w ℓ · · · w n , and C 2 b e the cycle induced b y w i w i +1 · · · w ℓ . F or each h ∈ [2], let G h b e the subgraph of G induced b y the v ertices on or inside of C h . By induction, we hav e disjoin t M -colorings φ 1 1 , φ 1 2 , φ 2 3 for G 1 that extend φ 0 1 , φ 0 2 , φ 0 3 . No w in G 2 , w e inherit from G 1 disjoin t M -colorings of G [ { w i , w ℓ } ]; denote them b y φ 00 1 , φ 00 2 , φ 00 3 . Again, by induction, G 2 has disjoin t M -colorings φ 2 1 , φ 2 2 , φ 2 3 , that extend φ 00 1 , φ 00 2 , φ 00 3 . Since φ 1 j and φ 2 j agree on w i , w ℓ , for each j ∈ [3], their unions give the desired disjoin t M -colorings φ j for G . No w we assume instead that the b oundary cycle C of the outer face has no chord. Our plan is to delete w n and remo ve certain colors from the lists of neighbors of w n , other than w 1 and w n − 1 . W e get the desired disjoint M -colorings for this smaller graph b y induction, and afterw ard sho w that we can extend these colorings to the desired colorings for G . No w we give more details. W e will partition [6] as R 1 ⊎ R 2 ⊎ R 3 suc h that, for all j ∈ [3], we hav e | R j | = 2 and R j ⊆ L j ( w n ) and M w 1 w n ( φ j ( w 1 )) / ∈ R j . (1) (Here R stands for “reserved” since these are colors in L j ( w n ) that we do not allo w to b e blo ck ed b y any neigh b or of w n , except for p ossibly w n − 1 .) W e first assume that w e can find such a partition of [6] satisfying (1), and use this partition to construct the disjoint M -colorings. Near the end of the pro of, we sho w how to find such a partition satisfying (1). Also, see Example 1. Let G ′ := G − w n . Let L ′ j ( v ) := L j ( v ) \ M w n v ( R j ) for eac h j ∈ [3] and each v ∈ N G ( w n ) \ { w 1 , w n − 1 } . F or all other v ∈ V ( G ′ ), let L ′ j ( v ) := L j ( v ) for all j ∈ [3]. No w we m ust find disjoint M -colorings φ ′ 1 , φ ′ 2 , φ ′ 3 for G ′ suc h that φ ′ j ( v ) ∈ L ′ j ( v ) for all v ∈ V ( G ′ ) and all j ∈ [3]. This essen tially holds by the induction hypothesis, but we remark on a few details. Because G has no c hords, each v ∈ V ( G ′ ) \ { w 1 , w 2 } has | L ′ j ( v ) | ⩾ 4 for all j ∈ [3] and ∪ 3 j =1 L ′ j ( v ) = [6]. Let φ ′ 1 , φ ′ 2 , φ ′ 3 denote the colorings of G ′ guaran teed b y the induction hypothesis. T o extend φ ′ j to w n , w e simply pick φ j ( w n ) from R j \ { M w n − 1 w n ( φ ′ j ( w n − 1 )) } . This cannot create a conflict with any neighbor of w n , since M w 1 w n ( φ ′ j ( w 1 )) / ∈ R j and by construction M w n x ( R j ) ∩ L ′ j ( x ) = ∅ for each x ∈ N G ( w n ) \ { w 1 , w n − 1 } . M w 1 w n ( φ 1 ( w 1 )) M w 1 w n ( φ 2 ( w 1 )) M w 1 w n ( φ 3 ( w 1 )) R 1 R 2 R 3 1 1 2 2 3 5 23 15 46 1 1 5 5 3 6 23 16 45 3 3 1 1 4 5 14 25 36 3 3 5 5 4 6 14 26 35 T able 1: Left: The 8 p ossibilities for ( M w 1 w n ( φ 1 ( w 1 )) , M w 1 w n ( φ 2 ( w 1 )) , M w 1 w n ( φ 3 ( w 1 ))), up to p erm uting colors. Right: Our 4 partitions of [6] as R 1 ⊎ R 2 ⊎ R 3 . Finally , we construct our partition of [6] as R 1 ⊎ R 2 ⊎ R 3 . By Condition (2) of the hypothesis, and by p ossibly renaming colors, we assume that L 1 ( w n ) = { 1 , 2 , 3 , 4 } , L 2 ( w n ) = { 1 , 2 , 5 , 6 } , and L 3 ( w n ) = { 3 , 4 , 5 , 6 } . W e assume that M w 1 w n ( φ j ( w 1 )) ∈ L j ( w n ) for each j ∈ [3]; doing so makes our task no easier. W e can sw ap the names of colors 1 and 2, and 3 and 4, and 5 and 6. So, b y symmetry , w e need only consider the 8 p ossibilities for ( M w 1 w n ( φ 1 ( w 1 )) , M w 1 w n ( φ 2 ( w 1 )) , M w 1 w n ( φ 3 ( w 1 ))) sho wn in T able 1. In each case w e use the corresp onding partition shown on the right. It is straigh tforward to c heck that this partition has the desired prop erties. This finishes the pro of. 6 W e used symmetry twice to simplify the pro of of the Key Lemma. This makes the pro of more compact, but harder to implement. F or concreteness, we work through the example in Figure 1. Example 1. Figure 1 only shows N [ w n ]. The rightmost 3 vertices on the left side are w n − 1 , w n , and w 1 . Eac h trap ezoid encloses the num b ers 1 , . . . , 6 for some vertex (with 1 at the narrow end and 6 at the wide end). The shading of a v ertex shows the M -colorings φ j for which the corresp onding color is av ailable. The first third of the circle (12:00-4:00) represen ts φ 1 , and the second and final thirds represen t, resp ectiv ely , φ 2 and φ 3 . So the figure shows that φ 1 ( w 1 ) = 1, φ 2 ( w 1 ) = 3, φ 3 ( w 1 ) = 5. (F or clarity , we omit the matching edges incident to other colors at w 1 , since they are irrelev ant.) F rom the figure, w e see that L 1 ( w n ) = { 1 , 2 , 4 , 6 } , L 2 ( w n ) = { 1 , 3 , 4 , 5 } , and L 3 ( w n ) = { 2 , 3 , 5 , 6 } . F or brevit y , henceforth we shorten L j ( w n ) to L j . In the pro of of the Key Lemma, w e assume by symmetry that these lists are { 1 , 2 , 3 , 4 } , { 1 , 2 , 5 , 6 } , and { 3 , 4 , 5 , 6 } . So we need a p erm utation π to justify this assumption. Let π := (1)(2364)(5) and let L j := { π ( h ) : h ∈ L j } . Note that L 1 = { 1 , 2 , 3 , 4 } , L 2 = { 1 , 2 , 5 , 6 } , and L 3 = { 3 , 4 , 5 , 6 } , as desired. The v ector of colors forbidden by w 1 from use on w n for colorings φ 1 , φ 2 , φ 3 is (4 , 1 , 6). But we are actually in terested in ( π (4) , π (1) , π (6)) = (2 , 1 , 4). W e lo ok for a corresp onding line in T able 1, but do not immediately see one. So w e need a p ermutation that sw aps one or more of the pairs 1 , 2 and 3 , 4 and 5 , 6. Let ρ := (12)(34)(5)(6). No w ( ρ (2) , ρ (1) , ρ (4)) = (1 , 2 , 3), whic h is the first line of the table. (Note that if w e let L j := { ρ ( h ) : h ∈ L j } , then L j = L j for all j ∈ [3].) Based on the table, w e let R 1 := { 2 , 3 } , R 2 := { 1 , 5 } , R 3 := { 4 , 6 } . No w let R j := { ρ − 1 ( h ) : h ∈ R j } . So R 1 = { 1 , 4 } , R 2 = { 2 , 5 } , R 3 = { 3 , 6 } . Finally , let R j := { π − 1 ( h ) : h ∈ R j } . So R 1 = { 1 , 6 } , w 1 w n − 1 w n w 1 w n − 1 Figure 1: Left: The p ortion of M induced by N [ w n ]. Right: The p ortion induced by N [ w n ] − { w n } . (Eac h thic k gra y line, say b et w een v ertices x and y , denotes a p erfect matc hing b etw een [6] x and [6] y . All other aspects of the figure are explained in Example 1.) 7 R 2 = { 4 , 5 } , R 3 = { 2 , 3 } . Finally , w e follow the matc hing edges from R 1 , R 2 , R 3 to see what colors are excluded from each L j at each neighbor of w n , when by the induction hypothesis we find the desired disjoint M -colorings for G − w n . T o finish this section, w e prov e part (ii) of our Main Theorem. Lemma 4. L et G := K 3 , 120 3 . Ther e exists a c orr esp ondenc e 5-c over M for G such that G do es not have 3 disjoint M -c olorings. This pr oves p art (ii) of the Main The or em, sinc e K 3 , 120 3 ∈ K 5 . Pr o of. Denote the parts of G b y U and W , where U = { u 1 , . . . , u 120 3 } and W = { x, y , z } . W e first pro ve the second statement. If G contains a K 5 -minor, then V ( G ) con tains disjoint subsets V 1 , . . . , V 5 that when contracted give V ( K 5 ). These sets are disjoint, so at least t w o con tain no v ertex of W . Th us their corresp onding v ertices in the con traction are non-adjacen t, a con tradiction. No w w e prov e the first statement. F or eac h v ∈ V ( G ), denote b y [5] v the set { 1 v , . . . , 5 v } of allo wable colors for v . Let M denote the set of all 5! p ossible p erfect matc hings b et ween [5] and [5]. W e now build our 5-cov er M of G . F or each triple ( M x , M y , M z ) ∈ M × M × M , c ho ose a unique v ertex u i in U , and include matching M x b et w een [5] x and [5] u i , matching M y b et w een [5] y and [5] u i , and matching M z b et w een [5] z and [5] u i . Supp ose, for a contradiction, that G admits 3 disjoin t M -colourings φ 1 , φ 2 , φ 3 . Here eac h φ i is a function from V ( G ) to [5]. Let N x := { ( φ 1 ( x ) , 1), ( φ 2 ( x ) , 2) , ( φ 3 ( x ) , 3) } and N y := { ( φ 1 ( y ) , 3), ( φ 2 ( y ) , 1) , ( φ 3 ( y ) , 2) } and N z := { ( φ 1 ( z ) , 2), ( φ 2 ( z ) , 3) , ( φ 3 ( z ) , 1) } . Cho ose a v ertex u i suc h that the matchings b et ween x and u i , y and u i , and z and u i are N x , N y , N z . Due to the three cov er-edges in the first co ordinate, w e m ust ha ve φ 1 ( u i ) ∈ { 4 , 5 } . Due to the second co ordinate, φ 2 ( u i ) ∈ { 4 , 5 } ; and due to the third co ordinate, φ 3 ( u i ) ∈ { 4 , 5 } . Therefore, φ 1 ( u i ) , φ 2 ( u i ) , φ 3 ( u i ) cannot b e disjoint, a con tradiction. W e observe that Theorem 4 is sharp as follows. F or every corresp ondence 5-co ver M , and every M -coloring φ 1 , there exists a disjoin t M -coloring φ 2 . This holds for every graph K 3 ,r b y induction on r , using an argumen t similar to (but simpler than) that pro ving Lemma A. In fact, the same is true of every graph that is 3-degenerate. F urthermore, Lemma 4 should also hold for K 3 ,t with t m uch smaller than 120 3 , and the ideas in [6] should help to achiev e suc h an optimization. Motiv ated b y Hadwiger’s Conjecture, we prop ose pro ving b etter b ounds on χ ⋆ c ( K s ) for all s ⩾ 6. Determining such v alues precisely for general s seems hard. (In particular, if they are linear in s , then a pro of would imply the famous Linear Hadwiger Conjecture: There exists a constan t a suc h that if G is K s -minor-free, then χ ( G ) ⩽ as .) But perhaps the v alues can be determined, or b ounded tigh tly , for some further small v alues of s . In this direction, w e men tion that χ ⋆ c ( K s ) > 2 s − 5 for all in tegers s . Sp ecifically , this is witnessed b y the complete bipartite graph K s − 2 ,r when we let r := ((2 s − 5)!) s − 2 . T o construct our (2 s − 5)- co ver M , we generalize the approach in the pro of of Theorem 4, taking all p ossible ordered sets of matc hings from the small side to the big side. It w as previously observ ed [4, Prop osition 24] that this cov er do es not admit 2 s − 5 disjoint M -colorings. In fact, the same argumen t in the previous pro of shows that it do es not even contain s − 2 disjoint M -colorings. W e end with a question. Question 5. L et G b e a K 5 -minor-fr e e gr aph. Do es every 5 -c over M of G admit two disjoint M -c olorings? 8 If the answer to Question 5 is yes, then it directly implies the first part of our Main Theorem, as from a 6-cov er M one can choose an arbitrary M -coloring φ , construct a 5-cov er M 2 from M b y remo ving φ ( v ) for every vertex v , and then find tw o disjoin t M 2 -colorings. On the other hand, if the answ er is no, then it will underscore that 6-co v ers are essen tial to obtain three disjoin t colorings. References [1] P . Bradshaw, I. Choi, and A. Kosto c hk a. Flexible DP 3-coloring of sparse m ultigraphs, 2025, . [2] S. Cambie and W. Cames v an Baten burg. F ractional list packing for lay ered graphs, 2024, . 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