(1,{lambda})-embedded graphs and the acyclic edge choosability

(1 , λ )-embedded graphs and the ac yclic edge choosability ∗ Xin Zhang † , Guizhen Liu, Jian-Liang W u School of Mathematics, Shandong Univ ersity , Jinan 250100, P . R. China Abstract A (1 , λ )-embedded graph is a graph that can be embedded on a surface with Euler characteristic λ so that each edge is crossed by at most one other edge. A graph G is called α -linear if there exists an integral constant β such that e ( G 0 ) ≤ αv ( G 0 ) + β for each G 0 ⊆ G . In this paper , it is sho wn that ev ery (1 , λ )-embedded graph G is 4-linear for all possible λ , and is acyclicly edge-(3 ∆ ( G ) + 70)-choosable for λ = 1 , 2. K eywor ds : (1 , λ )-embedded graph, α -linear graph, acyclic edge choosability . MSC : 05C10, 05C15. 1 Intr oduction and basic definitions In this paper , all graphs considered are finite, simple and undirected. Let G be a graph, we use V ( G ), E ( G ), δ ( G ) and ∆ ( G ) to denote the verte x set, the edge set, the minimum degree and the maximum degree of a graph G . Let e ( G ) = | E ( G ) | and v ( G ) = | V ( G ) | . Moreover , for embedded graph G (i.e., a graph that can be embedded on a surface), by F ( G ) we denote the face set of G . Let f ( G ) = | F ( G ) | . The girth g ( G ) of a graph G is the length of the shortest c ycle of G . A k -, k + - and k − -verte x (or face) is a vertex (or face) of degree k , at least k and at most k , respectiv ely . A graph G is called α -linear if there exists an integral constant β such that e ( G 0 ) ≤ αv ( G 0 ) + β for each G 0 ⊆ G . Furthermore, if β ≥ 0, then G is said to be α -nonnegati ve-linear; and if β < 0, then G is said to be α -negati ve-linear . For other undefined concepts we refer the reader to [ 3 ]. A mapping c from E ( G ) to the sets of colors { 1 , · · · , k } is called a proper edge- k -coloring of G provided that any two adjacent edges receiv e di ff erent colors. A proper edge- k -coloring c of G is called an acyclic edge- k -coloring of G if there are no bichromatic cycles in G under the coloring c . The smallest number of colors such that G has an acyclic edge coloring is called the acyclic edge chr omatic number of G , denoted by χ 0 a ( G ). A graph is said to be acyclic edge- f -choosable, whenev er we giv e a list L e of f ( e ) colors to each edge e ∈ E ( G ), there exists an acyclic Emails: sdu.zhang@yahoo.com.cn (X. Zhang), gzliu@sdu.edu.cn (G. Liu), jlwu@sdu.edu.cn (J.-L. W u). ∗ This research is partially supported by Graduate Independent Innov ation Foundation of Shandong Univ ersity (No. yzc10040) and National Natural Science Foundation of China (No. 10971121, 11026184, 61070230). † The first author is under the support from The Chinese Ministry of Education Prize for Academic Doctoral Fellows 1 edge- k -coloring of G , where each element is colored with a color from its own list. If | L e | = k for edge e ∈ E ( G ), we say that G is acyclicly edge- k -choosable. The minimum integer k such that G is acyclicly edge- k -choosable is called the acyclic edge choice number of G , denoted by χ 0 c ( G ). Acyclic coloring problem introduced in [ 8 ] has been extensi vely studied in many papers. One of the famous conjectures on the acyclic chromatic index is due to Alon, Sudako v and Zaks [ 2 ]. They conjectured that χ 0 a ( G ) ≤ ∆ ( G ) + 2 for any graph G . Alon et al.[ 1 ] prov ed that χ 0 a ( G ) ≤ 64 ∆ ( G ) for any graph G by using probabilistic arguments. This bound for arbitrary graph was later improved to 16 ∆ ( G ) by Molloy and Reed [ 9 ] and recently improv ed to 9 . 62 ∆ ( G ) by Ndreca et al.[ 10 ]. In 2008, Fiedoro wicz et al.[ 7 ] proved that χ 0 a ( G ) ≤ 2 ∆ ( G ) + 29 for each planar graph G by applying a combinatorial method. Now adays, acyclic coloring problem has attracted more and more attention since Coleman et al.[ 4 , 5 ] identified acyclic coloring as the model for computing a Hessian matrix via a substitution method. Thus to consider the acyclic coloring problems on some other special classes of graphs seems interesting. A graph is called 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. The notion of 1-planar -graph w as introduced by Ringel[ 11 ] while trying to simultaneously color the vertices and faces of a planar graph such that an y pair of adjacent / incident elements receiv e di ff erent colors. In f act, from a planar graph G , we can construct a 1-planar graph G 0 with its vertex set being V ( G ) ∪ F ( G ), and any two vertices of G 0 being adjacent if and only if their corresponding elements in G are adjacent or incident. Now we generalize this concept to (1 , λ )-embedded graph, namely , a graph that can be embedded on a surface S with Euler characteristic λ so that each edge is crossed by at most one other edge. Actually , a (1 , 2)-embedded graph is a 1-planar graph. It is sho wn in man y papers such as [ 6 ] that e ( G ) ≤ 4 v ( G ) − 8 for e very 1-planar graph G . Whereafter , to determine whether the number of edges in the class of (1 , λ )-embedded graphs is linear or not linear in the number of vertices for ev ery λ ≤ 2 might be interesting. In this paper , we first in vestigate some structures of (1 , λ )-embedded graph G in Section 2 and then gi ve a relation- ship among the three parameters e ( G ), v ( G ) and g ( G ) of G , which implies that e very (1 , λ )-embedded graph is 4-linear for any λ ≤ 2. In Section 3, we will introduce a linear upper bound for the acyclic edge choice number of the classes (1 , λ )-embedded graphs with special gi ven λ . 2 The linearity of ( 1 , λ )-embedded graphs Gi ven a ”good” graph G (i.e., one for which all intersecting edges intersect in a single point and arise from four distinct vertices), the crossing number , denoted by cr ( G ), is the minimum possible number of crossings with which the graph can be drawn. Let G be a (1 , λ )-embedded graph. In the following we always assume that G has been embedded on a surface with Euler characteristic λ so that each edge is crossed by at most one other edge and the number of crossings of G in this embedding is minimum. Thus, G has exactly cr ( G ) crossings. Sometimes we say such an embedding proper for con venience. Theorem 2.1. Let G be a (1 , λ )-embedded graph. Then cr ( G ) ≤ v ( G ) − λ . 2 Proof. Suppose G has been properly embedded on a surface with Euler characteristic λ . Then for each pair of edges ab , cd that cross each other at a crossing point s , their end vertices are pairwise distinct. For each such pair , we add ne w edges ac , cb , bd , d a (if it does not exist originally) to close s , then arbitrarily delete one edge ab or cd from G . Denote the resulting graph by G ∗ and then we ha ve cr ( G ∗ ) = 0. By Euler’ s formula v ( G ∗ ) − e ( G ∗ ) + f ( G ∗ ) = λ and the well-kno wn relation P v ∈ V ( G ∗ ) d V ( G ∗ ) ( v ) = P f ∈ F ( G ∗ ) d V ( G ∗ ) ( f ) = 2 e ( G ∗ ), f ( G ∗ ) ≤ 2 v ( G ∗ ) − 2 λ . Since each crossing point s (note that s is not a real vertex in G ) lies on a common boundary of two faces of G ∗ and each face of G ∗ is incident with at most one crossing point (recall the definition of G ∗ ), we deduce that 2 cr ( G ) ≤ f ( G ∗ ). Since v ( G ) = v ( G ∗ ), we hav e cr ( G ) ≤ f ( G ∗ ) 2 ≤ v ( G ∗ ) − λ = v ( G ) − λ in final.  Theorem 2.2. Let G be a (1 , λ )-embedded graph with girth at least g . Then e ( G ) ≤ 2 g − 2 g − 2 ( v ( G ) − λ ). Proof. Suppose G has been properly embedded on a surface with Euler characteristic λ . No w for each pair of edges ab , cd that cross each other , we arbitrarily delete one from G . Let G 0 be the resulting graph. One can easily see that cr ( G 0 ) = 0. By Euler’ s formula v ( G 0 ) − e ( G 0 ) + f ( G 0 ) = λ and the relations v ( G 0 ) = v ( G ), e ( G 0 ) = e ( G ) − cr ( G ), we hav e v ( G ) − e ( G ) + f ( G 0 ) = v ( G 0 ) − e ( G 0 ) + f ( G 0 ) − cr ( G ) = λ − cr ( G ) (2.1) and X f ∈ F ( G 0 ) d G 0 ( f ) = 2 e ( G 0 ) = 2( e ( G ) − cr ( G )) ≥ g ( G 0 ) f ( G 0 ) ≥ g ( G ) f ( G 0 ) ≥ g · f ( G 0 ) . (2.2) No w combine equations ( 2.1 ) and ( 2.2 ) together , we immediately ha ve e ( G ) ≤ g g − 2 ( v ( G ) − λ ) + cr ( G ) ≤ 2 g − 2 g − 2 ( v ( G ) − λ ) by Theorem 2.1 .  By Theorem 2.2 , the follo wing two corollaries are natural. Corollary 2.3. Ev ery (1 , λ )-embedded graph is 4-linear for any λ ≤ 2. Corollary 2.4. Ev ery triangle-free (1 , λ )-embedded graph is 3-linear for any λ ≤ 2. 3 Acyclic edge choosability of ( 1 , λ )-embedded graphs In this section we mainly in vestigate the acyclic edge choosability of (1 , λ )-embedded graphs with special gi ven λ . In [ 7 ], Fiedoro wicz et al. proved the follo wing two results. Theorem 3.1. If G is a graph such that e ( G 0 ) ≤ 2 v ( G 0 ) − 1 for each G 0 ⊆ G , then χ 0 a ( G ) ≤ ∆ ( G ) + 6. Theorem 3.2. If G is a graph such that e ( G 0 ) ≤ 3 v ( G 0 ) − 1 for each G 0 ⊆ G , then χ 0 a ( G ) ≤ 2 ∆ ( G ) + 29. In fact, these two theorems respectively imply that the acyclic edge chromatic number of 2-negati ve-linear graph G is at most ∆ ( G ) + 6 and that the acyclic edge chromatic number of 3-negati ve-linear graph G is at most 2 ∆ ( G ) + 29. Note that e very triangle-free (1 , λ )-embedded graph is 3-ne gati ve-linear for an y 1 ≤ λ ≤ 2 by Theorem 2.2 . Hence the follo wing corollary is tri vial. 3 Corollary 3.3. Let G be a triangle-free (1 , λ )-embedded graph with 1 ≤ λ ≤ 2. Then χ 0 a ( G ) ≤ 2 ∆ ( G ) + 29. The follo wing main theorem in this section is dedicated to gi ving a linear upper bound for the acyclic edge choice number of 4-negati ve-linear graphs. Theorem 3.4. If G is a graph such that e ( G 0 ) ≤ 4 v ( G 0 ) − 1 for each G 0 ⊆ G , then χ 0 c ( G ) ≤ 3 ∆ ( G ) + 70. As an immediately corollary of Theorems 2.2 and 3.4 , we hav e the following result. Corollary 3.5. Let G be a (1 , λ )-embedded graph with 1 ≤ λ ≤ 2. Then χ 0 c ( G ) ≤ 3 ∆ ( G ) + 70. Before proving Theorem 3.4 , we first sho w an useful structural lemma. Lemma 3.6. Let G be a graph such that e ( G ) ≤ 4 v ( G ) − 1 and δ ( G ) ≥ 4, Then at least one of the follo wing configura- tions occurs in G : ( C 1) a 4-verte x adjacent to a 19 − -verte x; ( C 2) a 5-verte x adjacent to two 19 − -vertices; ( C 3) a 6-verte x adjacent to four 19 − -vertices; ( C 4) a 7-verte x adjacent to six 19 − -vertices; ( C 5) a verte x v such that 20 ≤ d ( v ) ≤ 22 and at least d ( v ) − 3 of its neighbors are 7 − -vertices; ( C 6) a verte x v such that 23 ≤ d ( v ) ≤ 25 and at least d ( v ) − 2 of its neighbors are 7 − -vertices; ( C 7) a verte x v such that 26 ≤ d ( v ) ≤ 28 and at least d ( v ) − 1 of its neighbors are 7 − -vertices; ( C 8) a verte x v such that 29 ≤ d ( v ) ≤ 31 and all its neighbors are 7 − -vertices; ( C 9) a verte x v such that at least d ( v ) − 7 of its neighbors are 7 − -vertices and at least one of them is of de gree 4. Proof. Suppose, to the contrary , that none of the nine configurations occurs in G . W e assign to each v ertex v a char ge w ( v ) = d ( v ) − 8, then P v ∈ V ( G ) w ( v ) = P v ∈ V ( G ) ( d ( v ) − 8) ≤ − 2. In the following, we will reassign a new char ge denoted by w 0 ( x ) to each x ∈ V ( G ) according to some dischar ging rules. Since our rules only mo ve charges around, and do not a ff ect the sum, we hav e X v ∈ V ( G ) w 0 ( v ) = X v ∈ V ( G ) w ( v ) ≤ − 2 . (3.1) W e next show that w 0 ( v ) ≥ 0 for each v ∈ V ( G ), which leads to a desired contradiction. W e say a v ertex big (resp. small) if it is a 20 + -verte x (resp. 7 − -verte x). The discharging rules are defined as follo ws. ( R 1) Each big verte x gi ves 1 to each adjacent 4-v ertex. ( R 2) Each big verte x gi ves 3 4 to each adjacent verte x of degree between 5 and 7. Let v be a 4-verte x. Since ( C 1) does not occur , v is adjacent to four big vertices. So v totally receiv es 4 by ( R 1). This implies that w 0 ( v ) = w ( v ) + 4 = d ( v ) − 4 = 0. Similarly , we can also prov e the nonnegati vity of w 0 ( v ) if v is a k -vertex where 5 ≤ k ≤ 7. Let v be a k -vertex where 8 ≤ k ≤ 19. Since v is not in volved in the discharging rules, w 0 ( v ) = w ( v ) = d ( v ) − 8 ≥ 0. Let v be a k -vertex where 20 ≤ k ≤ 22. If v is adjacent to a 4-vertex, then v is adjacent to at most d ( v ) − 8 small vertices since ( C 9) does not occur . Since v sends each small v ertex at most 1 By ( R 1) and ( R 2), w 0 ( v ) ≥ w ( v ) − ( d ( v ) − 8) = 0. If v is adjacent to no 4-vertices, then v sends each small verte x 3 4 by ( R 2). Since ( C 5) 4 does not occur either , v is adjacent at most d ( v ) − 4 small vertices. So w 0 ( v ) ≥ w ( v ) − 3 4 ( d ( v ) − 4) = 1 4 ( d ( v ) − 20) ≥ 0. By similar arguments as abov e, we can also respecti vely show the nonnegati vity of w 0 ( v ) if v is a k -verte x where k ≥ 23.  Proof of Theorem 3 . 4 . Let K stands for 3 ∆ ( G ) + 70. W e prov e the theorem by contradiction. Let G be a counterex- ample to the theorem with the number of edges as small as possible. So there exists a list assignment L of K colors such that G is not acyclicly edge- L -choosable. For each coloring c of G , we define c ( u v ) to be the color of edge u v and set C ( u ) = { c ( u v ) | u v ∈ E ( G ) } for each vertex u . For W ⊆ V ( G ), set C ( W ) = S w ∈ W C ( w ). If u v ∈ E ( G ), we let W G ( v, u ) stands for the set of neighbors w of v in G such that c ( vw ) ∈ C ( u ). Now , we first prove that δ ( G ) ≥ 4. Suppose that there is a 3-vertex v ∈ V ( G ). Denote the three neighbors of v by x , y and z . Then by the minimality of G , the graph H = G − u x is acyclicly edge- L -choosable. Let c be an acyclic edge coloring of H . W e can extend c to u v by defining a list of av ailable colors for u v as follows: A ( u v ) = L ( u v ) \ ( C ( x ) ∪ C ( y ) ∪ C ( z )) . Since | C ( x ) | ≤ ∆ ( G ) − 1, | C ( y ) | ≤ ∆ ( G ) and | C ( z ) | ≤ ∆ ( G ), we have | A ( u v ) | ≥ K − 3 ∆ ( G ) + 1 > 0. So we can color u v by a color in A ( u v ) ⊆ L ( u v ), a contradiction. Similarly , one can also prove the absences of 1-vertices and 2-vertices in G . Hence δ ( G ) ≥ 4. Then by Lemma 3.6 , G contains at least one of the configurations ( C 1)-( C 9). In the following, we only show that if one of the configurations ( C 4), ( C 5) and ( C 9) appears, then we would get a contradiction. That is because the proofs are similar and easier for another six cases. Configuration ( C 4): Suppose that there is a 7-v ertex v who is adjacent to six 19 − -vertices, say x 1 , x 2 , · · · , x 6 . Denote another one neighbor of v by x 7 . Then by the minimality of G , the graph H = G − v x 7 is acyclicly edge- L -choosable. Let c be an acyclic edge coloring of H . Suppose c ( v x j ) < C ( x 7 ) for some 1 ≤ j ≤ 6. Then we can extend c to v x 7 by defining a list of av ailable colors for v x 7 as follo ws: A ( v x 7 ) = L ( v x 7 ) \ [ 1 ≤ i , j ≤ 7 C ( x i ) . Since | C ( x i ) | ≤ min { 19 , ∆ ( G ) } for e very 1 ≤ i ≤ 6 and | C ( x 7 ) | ≤ ∆ ( G ) − 1, we have | A ( v x 7 ) | ≥ K − min { ∆ ( G ) + 94 , 6 ∆ ( G ) − 1 } = max { 2 ∆ ( G ) − 24 , − 3 ∆ ( G ) + 69 } > 0. So we can color v x 7 by a color in A ( v x 7 ) ⊆ L ( v x 7 ), a contradiction. Thus we shall assume that c ( v x j ) ∈ C ( x 7 ) for ev ery 1 ≤ j ≤ 6. This implies that | S 1 ≤ i ≤ 7 C ( x i ) | ≤ ∆ ( G ) − 1 + 6 × min { 19 , ∆ ( G ) } − 6 = min { ∆ ( G ) + 107 , 7 ∆ ( G ) − 7 } . No w we extend c to v x 7 by defining a list of av ailable colors for v x 7 as follo ws: A ( v x 7 ) = L ( v x 7 ) \ [ 1 ≤ i ≤ 7 C ( x i ) . Note that | A ( v x 7 ) | ≥ K − min { ∆ ( G ) + 107 , 7 ∆ ( G ) − 7 } = max { 2 ∆ ( G ) − 37 , − 4 ∆ ( G ) + 77 } > 0. So we can again color v x 7 by a color in A ( v x 7 ) ⊆ L ( v x 7 ), also a contradiction. Configuration ( C 5): If there is a verte x v such that 20 ≤ d ( v ) ≤ 22 and at least d ( v ) − 3 of its neighbors are 7 − -vertices. W ithout loss of generality , we assume that d ( v ) = 22 and that v hav e nineteen 7 − -neighbors. Denote another three neighbors of v by x , y and z . Choose one 7 − -neighbor , say u , of v . Without loss of generality , we assume that d ( u ) = 7. 5 Then by the minimality of G , the graph H = G − u v is acyclicly edge- L -choosable. Let c be an acyclic edge coloring of H . Then we can extend c to u v by defining a list of av ailable colors for u v as follo ws: A ( u v ) = L ( u v ) \{ C ( u ) ∪ C ( v ) ∪ C ( x ) ∪ C ( y ) ∪ C ( z ) ∪ C ( W H ( v, u ) } . Since c is an acyclic (and thus it is proper), | W H ( v, u ) | ≤ d ( u ) − 1 = 6. Since | C ( x ) | ≤ ∆ ( G ), | C ( y ) | ≤ ∆ ( G ), | C ( z ) | ≤ ∆ ( G ) and | C ( w ) | ≤ 7 for each w ∈ W H ( v, u ), we ha ve | A ( u v ) | ≤ K − (3 ∆ ( G ) + 6 + 6 × 6 + 21 − 9) > 0. So we can color u v by a color in A ( u v ) ⊆ L ( u v ), a contradiction. Configuration ( C 9) If there is a vertex v such that at least d ( v ) − 7 of its neighbors are 7 − -vertices and at least one of them is of degree 4, say u . Denote another three neighbors of u by x , y and z . Let C 1 = { c ( vw ) | w ∈ N H ( v ) and d H ( w ) > 7 } and C 2 = { c ( vw ) | w ∈ N H ( v ) and d H ( w ) ≤ 7 } . Then | C 1 | ≤ 7. By the minimality of G , the graph H = G − u v is acyclicly edge- L -choosable. Let c be an acyclic edge coloring of H . Suppose C ( u ) ∩ C 1 , ∅ . W ithout loss of generality , we assume that c ( u x ) ∈ C 1 . No w we erase the color of the edge u x from c and recolor it from the list defined as follo ws: A ( u x ) = L ( u x ) \{ C ( x ) ∪ C ( y ) ∪ C ( z ) ∪ C 1 } . Since | C ( x ) | ≤ ∆ ( G ), | C ( y ) | ≤ ∆ ( G ), | C ( z ) | ≤ ∆ ( G ) and | C 1 | ≤ 7, we hav e | A ( u x ) | ≥ K − (3 ∆ ( G ) + 7) > 0. Note that A ( u x ) is just a sub-list of the original list given at the beginning of the proof and the new color of u x preserves the acyclicity of the coloring of H . So we can assume that C ( u ) ∩ C 1 = ∅ . In this case, we can extend c to the edge u v by defining a list of av ailable colors for u v as follo ws: A ( u v ) = L ( u v ) \{ C ( u ) ∪ C 1 ∪ C 2 ∪ C ( W H ( v, u )) } . Since C ( u ) ∩ C 1 = ∅ , we have c ( vw ) ∈ C 2 for each w ∈ W H ( v, u ) and thus | C ( W H ( v, u )) | ≤ 7 d H ( u ) = 21. Since | C 1 ∪ C 2 | = d H ( v ) ≤ ∆ ( G ) − 1 and | C ( u ) | = 3, we have | A ( u v ) | ≥ K − ( ∆ ( G ) + 23) > 0. So we can color u v by a color in A ( u v ) ⊆ L ( u v ). This contradiction completes the proof of Theorem 3.4 .  Refer ences [1] N. Alon, C. J. H. McDiarmid, B. A. Reed, Acyclic coloring of graphs, Random Structur es and Algorithms , 2, (1991), 277-288. [2] N. Alon, B. Sudakov , A. 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