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Upper bounds on adjacent vertex distinguishing total chromatic number of graphs
Discrete Applied Mathematics 233 (2017) 29–32
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Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Upper bounds on adjacent vertex distinguishing total chromatic number of graphs Xiaolan Hu a, *, Yunqing Zhang b , Zhengke Miao c a b c
School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, PR China Department of Mathematics, Nanjing University, Nanjing 210093, PR China School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu, 221116, PR China
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Article history: Received 29 March 2016 Received in revised form 22 July 2017 Accepted 14 August 2017 Available online 21 September 2017 Keywords: Adjacent vertex distinguishing total coloring Chromatic number Edge chromatic number Maximum degree
a b s t r a c t An adjacent vertex distinguishing total coloring of a graph G is a proper total coloring of G such that for any pair of adjacent vertices, the set of colors appearing on the vertex and incident edges are different. The minimum number of colors required for an adjacent vertex distinguishing total coloring of G is denoted by χa′′ (G). Let G be a graph, and χ (G) and χ ′ (G) be the chromatic number and edge chromatic number of G, respectively. In this paper we show that χa′′ (G) ≤ χ (G) + χ ′ (G) − 1 for any graph G with χ (G) ≥ 6, and χa′′ (G) ≤ χ (G) + ∆(G) for any graph G. Our results improve the only known upper bound 2∆ obtained by Huang et al. (2012). As a direct consequence, we have χa′′ (G) ≤ ∆(G) + 3 if χ (G) = 3 and thus it implies the known results on graphs with maximum degree 3, K4 -minor-free graphs, outerplanar graphs, graphs with maximum average degree less than 3, planar graphs with girth at least 4 and 2-degenerate graphs. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Let G be a graph with vertex set V (G) and edge set E(G). We use NG (v ) to denote the neighbors of a vertex v in G and dG (v ) = |NG (v )|. Let ∆(G) denote the maximum degree of G. If there is no confusion from the context we use simply ∆. A proper k-coloring is a mapping φ : V (G) −→ {1, 2, . . . , k} such that any two adjacent vertices in V (G) receive different colors. The chromatic number χ (G) of G is the smallest integer k such that G has a proper k-coloring. A proper k-edge coloring is a mapping φ : E(G) −→ {1, 2, . . . , k} such that any two adjacent edges in E(G) receive different colors. The edge chromatic number χ ′ (G) of G is the smallest integer k such that G has a proper k-edge coloring. A proper total k-coloring is a mapping φ : V (G) ∪ E(G) −→ {1, 2, . . . , k} such that any two adjacent or incident elements in V (G) ∪ E(G) receive different colors. The total chromatic number χ ′′ (G) of G is the smallest integer k such that G has a proper total k-coloring. Let φ be a proper total coloring of G. Denote Cφ (v ) = {φ (uv ) | uv ∈ E(G)}∪{φ (v )}. A proper total k-coloring φ of G is adjacent vertex distinguishing, or a total-k-avd-coloring, if Cφ (u) ̸ = Cφ (v ) whenever uv ∈ E(G). The adjacent vertex distinguishing total chromatic number χa′′ (G) is the smallest integer k such that G has a total-k-avd-coloring. It is obvious that ∆ + 1 ≤ χ ′′ (G) ≤ χa′′ (G). In [13], Zhang et al. introduced the notation of adjacent vertex distinguishing total colorings of graphs and made the following conjecture in terms of the maximum degree ∆. Conjecture 1.1. For any graph G, χa′′ (G) ≤ ∆ + 3.
*
Corresponding author. E-mail addresses: [email protected] (X. Hu), [email protected] (Y. Zhang), [email protected] (Z. Miao).
http://dx.doi.org/10.1016/j.dam.2017.08.016 0166-218X/© 2017 Elsevier B.V. All rights reserved.
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Chen [2] and Wang [8], independently, confirmed Conjecture 1.1 for graphs G with ∆ ≤ 3. Later, Hulgan [5] presented a more concise proof on this result. Wang et al. considered the adjacent vertex distinguishing total chromatic number of K4 -minor-free graphs [11] and outerplanar graphs [12]. In [10], Wang considered the adjacent vertex distinguishing total chromatic number of sparse graphs. Wang and Huang [9] considered the adjacent vertex distinguishing total colorings of planar graphs with ∆ ≥ 11. In [6], Miao et al. considered the adjacent vertex distinguishing total chromatic number of 2-degenerate graphs. For general graphs, Huang et al. [4] proved the following upper bound. Theorem 1.2. Let G be a graph with ∆ ≥ 3, then χa′′ (G) ≤ 2∆. By definitions of coloring, edge coloring and adjacent vertex distinguishing total coloring, we have the following proposition which is also observed in [4]: Proposition 1.3. For any graph G, χa′′ (G) ≤ χ (G) + χ ′ (G). In this paper, by applying recoloring techniques, we prove a new upper bound for χa′′ (G) for general graphs. Theorem 1.4. Let G be a graph with χ (G) ≥ 6. Then χa′′ (G) ≤ χ (G) + χ ′ (G) − 1. In this paper we first present a new upper bound on χa′′ (G). Theorem 1.5. For each graph G, χa′′ (G) ≤ χ (G) + ∆. Theorem 1.5 implies the upper bound 2∆ as stated in Theorem 1.2 as well as the case when ∆ = 3. By Four Color Theorem [1], it implies that for any planar graph G, χa′′ (G) ≤ ∆ + 4 while the conjecture is χa′′ (G) ≤ ∆ + 3. It also implies that Conjecture 1.1 is true for all 3-colorable graphs. Corollary 1.6. If G is 3-colorable, then χa′′ (G) ≤ ∆ + 3. Note that the family of 3-colorable graphs includes many interesting subfamilies of graphs such as graphs with maximum degree 3 (except K4 ), K4 -minor-free graphs, outerplanar graphs, graphs with maximum average degree less than 3, and 2degenerate graphs. Therefore Corollary 1.6 implies the known results for those families of graphs. Also by the well-known Grötzsch’s 3-color theorem [3], we have the following result which is a special case of Corollary 1.6. Corollary 1.7. Let G be a triangle-free planar graph. Then χa′′ (G) ≤ ∆ + 3 and thus Conjecture 1.1 is true for all triangle-free planar graphs. 2. Proof of Theorem 1.5 Proof. Denote k = χ (G). Let φ : V (G) → {∆ + 1, ∆ + 2, . . . , ∆ + k} be a proper vertex coloring of G. By Vizing’s Theorem [7], every graph G is (∆ + 1)-edge coloring. Let ψ : E(G) → {1, 2, . . . , ∆ + 1} be a proper edge coloring of G. Then φ together with ψ is a total coloring which may not be proper since some edges colored with ∆ + 1 may be incident with a vertex colored with ∆ + 1 too. Let A be the set of all edges which are colored with ∆ + 1 and are incident with a vertex colored with ∆ + 1. That is A = {uv ∈ E(G) | ψ (uv ) = ∆ + 1 ∈ {φ (u), φ (v )}}. Then A is a matching since ψ is a proper edge coloring. Denote V1 = {u | φ (u) = ∆ + 1, and u is incident with an edge in A}. The V1 is an independent set. Let u ∈ V1 with uv ∈ A. Since ψ (uv ) = ∆ + 1 and d(u) ≤ ∆, there exists a color α ∈ {1, 2, . . . , ∆} \ {ψ (ux) | x ∈ N(u)}. Recolor u with α . Let ϕ be the new total coloring after applying the recoloring on all the vertices in V1 . Noting that the colors of the edges are unchanged and in {1, 2, . . . , ∆ + 1}, and the colors of the vertices not in V1 are unchanged and in {∆ + 1, ∆ + 2, . . . , ∆ + k}, we have ϕ is a (∆ + k)-proper total coloring. In the following, we show that ϕ is an adjacent vertex distinguishing (∆ + k)-total coloring. Let xy be an edge in G. Since V1 is an independent set, assume without loss of generality that y ̸ ∈ V1 and ϕ (y) = ∆ + j for some j ≥ 1. If x ∈ V1 , then φ (x) = ∆ + 1 and thus φ (y) = ∆ + j ≥ ∆ + 2. Now in the coloring ϕ , ϕ (x) < ∆ + 1. And thus Cϕ (x) ⊆ {1, 2, . . . , ∆ + 1} since the colors of all edges are unchanged and are in {1, 2, . . . , ∆ + 1}. Since ∆ + j = ϕ (y) ∈ Cϕ (y), then Cϕ (x) ̸ = Cϕ (y). If x ̸ ∈ V1 , then ϕ (u) = φ (u) = ∆ + i for some i ≥ 1 and i ̸ = j. We may assume without loss of generality that j ≥ 2. In this case Cϕ (x) = Cφ (x) and Cϕ (y) = Cφ (y). Since all the edges incident with x or y are colored with colors in {1, 2, . . . , ∆ + 1}, then ∆ + j ̸ ∈ Cϕ (x), and thus Cϕ (x) ̸ = Cϕ (y). Therefore, we show that Cϕ (x) ̸ = Cϕ (y) for any xy ∈ E(G), i.e., ϕ is a total-(k + ∆)-avd-coloring and therefore completes the proof of Theorem 1.5. ■
X. Hu et al. / Discrete Applied Mathematics 233 (2017) 29–32
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3. Proof of Theorem 1.4 Let b = χ (G) and a = χ ′ (G). Let φ1 : V (G) → {a, a + 1, . . . , a + b − 1} be a proper vertex coloring of G and φ2 : E(G) → {1, 2, . . . , a} be a proper edge coloring of G. Then the vertex coloring φ1 and the edge coloring φ2 together give a total coloring of G which may not be proper because we may have edges uv with a = φ2 (uv ) ∈ {φ1 (u), φ1 (v )}. Let A denote the set of all such edges. That is A = {uv ∈ E(G) | φ2 (uv ) = a ∈ {φ1 (u), φ1 (v )}}. Then A is a matching since all edges in A are colored with the color a in the proper edge coloring φ2 . For each edge uv ∈ A, suppose φ1 (u) = a and φ1 (v ) = a + i for some 1 ≤ i ≤ b − 1. Recolor the edge uv with φ1 (v ) + 1 = a + i + 1 (a + b = a + 1). Let φ be the new total coloring after recoloring all the edges in A. Claim 3.1. φ is a (a + b − 1)-proper total coloring of G. Proof. Since the colors of the vertices are unchanged and in {a, a + 1, . . . , a + b − 1}, and the colors of the edges not in A are unchanged and in {1, 2, . . . , a}, then φ is a (a + b − 1)-proper total coloring of G. ■ The following claim simply follows from the definitions of the coloring φ and A. Claim 3.2. In the coloring φ , e = uv ∈ A if and only if φ (uv ) ∈ {a + 1, a + 2, . . . , a + b − 1} and a ∈ {φ (u), φ (v )}. Moreover, if φ (u) = a and φ (v ) = a + i, then φ (uv ) = a + i + 1. Let B1 denote the set of edges uv with Cφ (u) = Cφ (v ) and B ⊆ B1 be a maximal matching. Claim 3.3. In the coloring φ , each edge in B is adjacent to an edge in A and their common endvertex is colored with the color a. Proof. Let uv ∈ B. Then {φ (u), φ (v )} ⊆ {a, a + 1, . . . , a + b − 1}. Assume without loss of generality that φ (v ) = a + j for some j ≥ 1. Since uv ∈ B, we have a + j ∈ Cφ (v ) = Cφ (u). Since φ (u) ̸ = φ (v ) = a + j, there is an edge uu1 colored with a + j. By Claim 3.2, uu1 ∈ A. Now we need to show a = φ (u). Otherwise, assume φ (u) = a + i, where i ≥ 1 and i ̸ = j. Thus a + i ∈ Cφ (u) = Cφ (v ). Since v is colored with a + j, there must be an edge vv1 colored with a + i. By Claim 3.2, we have φ (u1 ) = φ (v1 ) = a, a + j = φ (uu1 ) = φ (u) + 1 = a + i + 1, and a + i = φ (vv1 ) = φ (v ) + 1 = a + j + 1. Therefore a + i = a + i + 2. Since b ≥ 6, b − 1 ≥ 5. a + i ̸ = a + i + 2 (mod b − 1). This contradiction shows φ (u) = a and thus completes the proof of the claim. ■ For each edge uv ∈ B with φ (u) = a, φ (uv ) = t and φ (v ) = a + i for some 1 ≤ i ≤ b − 1, we recolor u with t and uv with φ (v ) + 2 = a + i + 2 (a + b = a + 1 and a + b + 1 = a + 2). Denote the new coloring ψ . The following two claims simply follow from the definitions of the coloring ψ , A and B. Claim 3.4. (1) For each edge uv , ψ (uv ) = φ (uv ) for each edge uv ̸ ∈ B. (2) For each vertex x ∈ V (G), ψ (x) = φ (x) unless x is incident with an edge xv ∈ B and φ (x) = a. In this case, ψ (x) = t ∈ {1, 2, . . . , a − 1}, ψ (xv ) = φ (v ) + 2, ψ (v ) = φ (v ) = a + j for some j ≥ 1. Moreover, t ̸∈ Cψ (v ) and thus Cψ (x) ̸ = Cψ (v ). Claim 3.5. In the coloring ψ , (1) uv ∈ B with ψ (u) < ψ (v ) if and only if ψ (u) ∈ {1, 2, . . . , a − 1}, ψ (v ) = a + j for some j ≥ 1 and ψ (uv ) = ψ (v ) + 2 = a + j + 2. (2) uv ∈ A with ψ (u) < ψ (v ) if and only if ψ (u) = a, ψ (v ) = a + j for some j ≥ 1 and ψ (uv ) = ψ (v ) + 1 = a + j + 1. (3) If uv ̸ ∈ A ∪ B with ψ (u) < ψ (v ), then ψ (uv ) = φ (uv ) ∈ {1, 2, . . . , a}, ψ (u) = φ (u) and ψ (v ) = φ (v ) = a + j for some j ≥ 1. By Claims 3.4 and 3.5, we have the following claim. Claim 3.6. ψ is a proper (a + b − 1)-total coloring of G. Claim 3.7. If uv is an edge in G with Cψ (u) = Cψ (v ), then uv ∈ A. Proof. Let uv be an edge with ψ (u) < ψ (v ) and Cψ (u) = Cψ (v ). Then ψ (v ) = φ (v ) = a + j for some j ≥ 1. By Claim 3.4, uv ̸ ∈ B. Suppose uv ̸ ∈ A. Then ψ (uv ) = φ (uv ) by the definitions of A, B and ψ, φ . We consider two cases depending on whether ψ (u) ≤ a or ψ (u) > a. If ψ (u) > a, we may assume that ψ (u) = a + i, then i < j. Since Cψ (u) = Cψ (v ), there must be an edge vv1 colored with a + i and an edge uu1 colored with a + j. By Claim 3.5, we have vv1 , uu1 ∈ A ∪ B. If vv1 ∈ A, then ψ (v1 ) = a, a + i = ψ (vv1 ) = ψ (v ) + 1 = a + j + 1, and if vv1 ∈ B, then ψ (v1 ) < a, a + i = ψ (vv1 ) = ψ (v ) + 2 = a + j + 2. If uu1 ∈ A,
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then ψ (u1 ) = a, a + j = ψ (uu1 ) = ψ (u) + 1 = a + i + 1, and if uu1 ∈ B, then ψ (u1 ) < a, a + j = ψ (uu1 ) = ψ (u) + 2 = a + i + 2. Therefore, a + i = a + i + 2 if vv1 , uu1 ∈ A, a + i = a + i + 3 if vv1 ∈ A, uu1 ∈ B or vv1 ∈ B, uu1 ∈ A, and a + i = a + i + 4 if vv1 , uu1 ∈ B. Since b ≥ 6, b−1 ≥ 5, then a+i ̸= a+i+2 (mod b−1), a+i ̸= a+i+3 (mod b−1), a+i ̸= a+i+4 (mod b−1), a contradiction. Now we assume ψ (u) ≤ a. Since v is colored with a + j and Cψ (u) = Cψ (v ), there is an edge uu1 colored with a + j. Then uu1 ∈ A ∪ B by Claim 3.5. If uu1 ∈ B, then ψ (u) < a, ψ (u1 ) ≥ a + 1, φ (u) = a and ψ (uu1 ) = ψ (u1 ) + 2. By Claim 3.3, there must be an edge uu2 ∈ A with ψ (uu2 ) = ψ (u1 ). Since ψ (uu2 ) ∈ Cψ (u) = Cψ (v ), there must be an edge vv1 colored with ψ (uu2 ). Then vv1 ∈ A ∪ B by Claim 3.5. If vv1 ∈ A, then ψ (vv1 ) = ψ (v ) + 1 = a + j + 1, and if vv1 ∈ B, then ψ (vv1 ) = ψ (v ) + 2 = a + j + 2. Noting that a + j = ψ (uu1 ) = ψ (u1 ) + 2 = ψ (uu2 ) + 2 = ψ (vv1 ) + 2, we have a + j = a + j + 3 if vv1 ∈ A, and a + j = a + j + 4 if vv1 ∈ B. Since b ≥ 6, b − 1 ≥ 5, then a + j ̸ = a + j + 3 (mod b − 1), a + j ̸ = a + j + 4 (mod b − 1), a contradiction. Therefore uu1 ∈ A. Then ψ (u) = a and φ (uu1 ) = a by Claim 3.5. Since a ∈ Cψ (u) = Cψ (v ), there must be an edge vv1 colored with a. Noting that uv ̸ ∈ B, there is an edge e adjacent to u or v such that e ∈ A ∪ B. Since ψ (e) ∈ Cψ (u) = Cψ (v ), there must be an edge uu2 colored with ψ (e), an edge vv2 colored with ψ (e), then uu2 , vv2 ∈ A ∪ B. Noting that uu1 ∈ A and A is a matching, we have uu2 ∈ B, then by Claim 3.5, ψ (u) < a, a contradiction. ■ Claim 3.8. ψ is a total (a + b − 1)-avd-coloring. Proof. If there is uv ∈ E(G) such that Cψ (u) = Cψ (v ), then uv ∈ A by Claim 3.7. By Claim 3.5, assume without loss of generality that ψ (u) = a, ψ (v ) = a + j for some j ≥ 1, then φ (uv ) = a and ψ (uv ) = ψ (v ) + 1 = a + j + 1. Since v is colored with a + j and Cψ (u) = Cψ (v ), there must be an edge uu1 colored with a + j. Then uu1 ∈ A ∪ B. Noting that uv ∈ A and A is a matching, we have uu1 ∈ B, then by Claim 3.5, ψ (u) < a, a contradiction. Therefore, we show that Cϕ (u) ̸ = Cϕ (v ) for any uv ∈ E(G), i.e., ϕ is a total-(a + b − 1)-avd-coloring and therefore completes the proof of Theorem 1.4. ■ Acknowledgments The first author’s research was partially supported by NSFC (No. 11601176) and NSF of Hubei Province (No. 2016CFB146). The second author’s research was partially supported by NSFC (No. 11671198). The third author’s research was partially supported by NSFC (Nos. 11571149 and 11571168). References [1] K. Appel, W. Haken, Every planar map is four colorable, Bull. Amer. Math. Soc. 82 (1976) 711–712. [2] X. Chen, On the adjacent vertex distinguishing total coloring numbers of graphs with ∆ = 3, Discrete Math. 308 (17) (2008) 4003–4007. [3] H. Grötzsch, Zur Theorie der diskreten Gebilde. VII. Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel. (German), Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg. Math.-Natur. Reih. 8 (1958/1959) 109–120. [4] D. Huang, W. Wang, C. Yan, A note on the adjacent vertex distinguishing total chromatic number of graphs, Discrete Math. 312 (24) (2012) 3544–3546. [5] J. Hulgan, Concise proofs for adjacent vertex-distinguishing total colorings, Discrete Math. 309 (2009) 2548–2550. [6] Z. Miao, R. Shi, X. Hu, R. Luo, Adjacent vertex distinguishing total colorings of 2-degenerate graphs, Discrete Math. 339 (2016) 2446–2449. [7] V. Vizing, On an estimate of the chromatic index of a p-graph, Diskret. Anal. 3 (1964) 25–30. [8] H. Wang, On the adjacent vertex distinguishing total chromatic number of the graphs with ∆ = 3, J. Comb. Optim. 14 (2007) 87–109. [9] W. Wang, D. Huang, The adjacent vertex distinguishing total coloring of planar graphs, J. Comb. Optim. 27 (2) (2014) 379–396. [10] W. Wang, Y. Wang, Adjacent vertex distinguishing total coloring of graphs with lower average degree, Taiwanese J. Math. 12 (4) (2008) 979–990. [11] W. Wang, P. Wang, On adjacent-vertex-distinguishing total coloring of K4 -minor-free graphs, Sci. China Ser. A 39 (12) (2009) 1462–1472 (in Chinese). [12] Y. Wang, W. Wang, Adjacent vertex distinguishing total colorings of outerplanar graphs, J. Comb. Optim. 19 (2010) 123–133. [13] Z. Zhang, X. Chen, J. Li, B. Yao, X. Lu, J. Wang, On adjacent-vertex-distinguishing total coloring of graphs, Sci. China Ser. A 48 (2005) 289–299.
Contents lists available at ScienceDirect
Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Upper bounds on adjacent vertex distinguishing total chromatic number of graphs Xiaolan Hu a, *, Yunqing Zhang b , Zhengke Miao c a b c
School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, PR China Department of Mathematics, Nanjing University, Nanjing 210093, PR China School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu, 221116, PR China
article
info
Article history: Received 29 March 2016 Received in revised form 22 July 2017 Accepted 14 August 2017 Available online 21 September 2017 Keywords: Adjacent vertex distinguishing total coloring Chromatic number Edge chromatic number Maximum degree
a b s t r a c t An adjacent vertex distinguishing total coloring of a graph G is a proper total coloring of G such that for any pair of adjacent vertices, the set of colors appearing on the vertex and incident edges are different. The minimum number of colors required for an adjacent vertex distinguishing total coloring of G is denoted by χa′′ (G). Let G be a graph, and χ (G) and χ ′ (G) be the chromatic number and edge chromatic number of G, respectively. In this paper we show that χa′′ (G) ≤ χ (G) + χ ′ (G) − 1 for any graph G with χ (G) ≥ 6, and χa′′ (G) ≤ χ (G) + ∆(G) for any graph G. Our results improve the only known upper bound 2∆ obtained by Huang et al. (2012). As a direct consequence, we have χa′′ (G) ≤ ∆(G) + 3 if χ (G) = 3 and thus it implies the known results on graphs with maximum degree 3, K4 -minor-free graphs, outerplanar graphs, graphs with maximum average degree less than 3, planar graphs with girth at least 4 and 2-degenerate graphs. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Let G be a graph with vertex set V (G) and edge set E(G). We use NG (v ) to denote the neighbors of a vertex v in G and dG (v ) = |NG (v )|. Let ∆(G) denote the maximum degree of G. If there is no confusion from the context we use simply ∆. A proper k-coloring is a mapping φ : V (G) −→ {1, 2, . . . , k} such that any two adjacent vertices in V (G) receive different colors. The chromatic number χ (G) of G is the smallest integer k such that G has a proper k-coloring. A proper k-edge coloring is a mapping φ : E(G) −→ {1, 2, . . . , k} such that any two adjacent edges in E(G) receive different colors. The edge chromatic number χ ′ (G) of G is the smallest integer k such that G has a proper k-edge coloring. A proper total k-coloring is a mapping φ : V (G) ∪ E(G) −→ {1, 2, . . . , k} such that any two adjacent or incident elements in V (G) ∪ E(G) receive different colors. The total chromatic number χ ′′ (G) of G is the smallest integer k such that G has a proper total k-coloring. Let φ be a proper total coloring of G. Denote Cφ (v ) = {φ (uv ) | uv ∈ E(G)}∪{φ (v )}. A proper total k-coloring φ of G is adjacent vertex distinguishing, or a total-k-avd-coloring, if Cφ (u) ̸ = Cφ (v ) whenever uv ∈ E(G). The adjacent vertex distinguishing total chromatic number χa′′ (G) is the smallest integer k such that G has a total-k-avd-coloring. It is obvious that ∆ + 1 ≤ χ ′′ (G) ≤ χa′′ (G). In [13], Zhang et al. introduced the notation of adjacent vertex distinguishing total colorings of graphs and made the following conjecture in terms of the maximum degree ∆. Conjecture 1.1. For any graph G, χa′′ (G) ≤ ∆ + 3.
*
Corresponding author. E-mail addresses: [email protected] (X. Hu), [email protected] (Y. Zhang), [email protected] (Z. Miao).
http://dx.doi.org/10.1016/j.dam.2017.08.016 0166-218X/© 2017 Elsevier B.V. All rights reserved.
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Chen [2] and Wang [8], independently, confirmed Conjecture 1.1 for graphs G with ∆ ≤ 3. Later, Hulgan [5] presented a more concise proof on this result. Wang et al. considered the adjacent vertex distinguishing total chromatic number of K4 -minor-free graphs [11] and outerplanar graphs [12]. In [10], Wang considered the adjacent vertex distinguishing total chromatic number of sparse graphs. Wang and Huang [9] considered the adjacent vertex distinguishing total colorings of planar graphs with ∆ ≥ 11. In [6], Miao et al. considered the adjacent vertex distinguishing total chromatic number of 2-degenerate graphs. For general graphs, Huang et al. [4] proved the following upper bound. Theorem 1.2. Let G be a graph with ∆ ≥ 3, then χa′′ (G) ≤ 2∆. By definitions of coloring, edge coloring and adjacent vertex distinguishing total coloring, we have the following proposition which is also observed in [4]: Proposition 1.3. For any graph G, χa′′ (G) ≤ χ (G) + χ ′ (G). In this paper, by applying recoloring techniques, we prove a new upper bound for χa′′ (G) for general graphs. Theorem 1.4. Let G be a graph with χ (G) ≥ 6. Then χa′′ (G) ≤ χ (G) + χ ′ (G) − 1. In this paper we first present a new upper bound on χa′′ (G). Theorem 1.5. For each graph G, χa′′ (G) ≤ χ (G) + ∆. Theorem 1.5 implies the upper bound 2∆ as stated in Theorem 1.2 as well as the case when ∆ = 3. By Four Color Theorem [1], it implies that for any planar graph G, χa′′ (G) ≤ ∆ + 4 while the conjecture is χa′′ (G) ≤ ∆ + 3. It also implies that Conjecture 1.1 is true for all 3-colorable graphs. Corollary 1.6. If G is 3-colorable, then χa′′ (G) ≤ ∆ + 3. Note that the family of 3-colorable graphs includes many interesting subfamilies of graphs such as graphs with maximum degree 3 (except K4 ), K4 -minor-free graphs, outerplanar graphs, graphs with maximum average degree less than 3, and 2degenerate graphs. Therefore Corollary 1.6 implies the known results for those families of graphs. Also by the well-known Grötzsch’s 3-color theorem [3], we have the following result which is a special case of Corollary 1.6. Corollary 1.7. Let G be a triangle-free planar graph. Then χa′′ (G) ≤ ∆ + 3 and thus Conjecture 1.1 is true for all triangle-free planar graphs. 2. Proof of Theorem 1.5 Proof. Denote k = χ (G). Let φ : V (G) → {∆ + 1, ∆ + 2, . . . , ∆ + k} be a proper vertex coloring of G. By Vizing’s Theorem [7], every graph G is (∆ + 1)-edge coloring. Let ψ : E(G) → {1, 2, . . . , ∆ + 1} be a proper edge coloring of G. Then φ together with ψ is a total coloring which may not be proper since some edges colored with ∆ + 1 may be incident with a vertex colored with ∆ + 1 too. Let A be the set of all edges which are colored with ∆ + 1 and are incident with a vertex colored with ∆ + 1. That is A = {uv ∈ E(G) | ψ (uv ) = ∆ + 1 ∈ {φ (u), φ (v )}}. Then A is a matching since ψ is a proper edge coloring. Denote V1 = {u | φ (u) = ∆ + 1, and u is incident with an edge in A}. The V1 is an independent set. Let u ∈ V1 with uv ∈ A. Since ψ (uv ) = ∆ + 1 and d(u) ≤ ∆, there exists a color α ∈ {1, 2, . . . , ∆} \ {ψ (ux) | x ∈ N(u)}. Recolor u with α . Let ϕ be the new total coloring after applying the recoloring on all the vertices in V1 . Noting that the colors of the edges are unchanged and in {1, 2, . . . , ∆ + 1}, and the colors of the vertices not in V1 are unchanged and in {∆ + 1, ∆ + 2, . . . , ∆ + k}, we have ϕ is a (∆ + k)-proper total coloring. In the following, we show that ϕ is an adjacent vertex distinguishing (∆ + k)-total coloring. Let xy be an edge in G. Since V1 is an independent set, assume without loss of generality that y ̸ ∈ V1 and ϕ (y) = ∆ + j for some j ≥ 1. If x ∈ V1 , then φ (x) = ∆ + 1 and thus φ (y) = ∆ + j ≥ ∆ + 2. Now in the coloring ϕ , ϕ (x) < ∆ + 1. And thus Cϕ (x) ⊆ {1, 2, . . . , ∆ + 1} since the colors of all edges are unchanged and are in {1, 2, . . . , ∆ + 1}. Since ∆ + j = ϕ (y) ∈ Cϕ (y), then Cϕ (x) ̸ = Cϕ (y). If x ̸ ∈ V1 , then ϕ (u) = φ (u) = ∆ + i for some i ≥ 1 and i ̸ = j. We may assume without loss of generality that j ≥ 2. In this case Cϕ (x) = Cφ (x) and Cϕ (y) = Cφ (y). Since all the edges incident with x or y are colored with colors in {1, 2, . . . , ∆ + 1}, then ∆ + j ̸ ∈ Cϕ (x), and thus Cϕ (x) ̸ = Cϕ (y). Therefore, we show that Cϕ (x) ̸ = Cϕ (y) for any xy ∈ E(G), i.e., ϕ is a total-(k + ∆)-avd-coloring and therefore completes the proof of Theorem 1.5. ■
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3. Proof of Theorem 1.4 Let b = χ (G) and a = χ ′ (G). Let φ1 : V (G) → {a, a + 1, . . . , a + b − 1} be a proper vertex coloring of G and φ2 : E(G) → {1, 2, . . . , a} be a proper edge coloring of G. Then the vertex coloring φ1 and the edge coloring φ2 together give a total coloring of G which may not be proper because we may have edges uv with a = φ2 (uv ) ∈ {φ1 (u), φ1 (v )}. Let A denote the set of all such edges. That is A = {uv ∈ E(G) | φ2 (uv ) = a ∈ {φ1 (u), φ1 (v )}}. Then A is a matching since all edges in A are colored with the color a in the proper edge coloring φ2 . For each edge uv ∈ A, suppose φ1 (u) = a and φ1 (v ) = a + i for some 1 ≤ i ≤ b − 1. Recolor the edge uv with φ1 (v ) + 1 = a + i + 1 (a + b = a + 1). Let φ be the new total coloring after recoloring all the edges in A. Claim 3.1. φ is a (a + b − 1)-proper total coloring of G. Proof. Since the colors of the vertices are unchanged and in {a, a + 1, . . . , a + b − 1}, and the colors of the edges not in A are unchanged and in {1, 2, . . . , a}, then φ is a (a + b − 1)-proper total coloring of G. ■ The following claim simply follows from the definitions of the coloring φ and A. Claim 3.2. In the coloring φ , e = uv ∈ A if and only if φ (uv ) ∈ {a + 1, a + 2, . . . , a + b − 1} and a ∈ {φ (u), φ (v )}. Moreover, if φ (u) = a and φ (v ) = a + i, then φ (uv ) = a + i + 1. Let B1 denote the set of edges uv with Cφ (u) = Cφ (v ) and B ⊆ B1 be a maximal matching. Claim 3.3. In the coloring φ , each edge in B is adjacent to an edge in A and their common endvertex is colored with the color a. Proof. Let uv ∈ B. Then {φ (u), φ (v )} ⊆ {a, a + 1, . . . , a + b − 1}. Assume without loss of generality that φ (v ) = a + j for some j ≥ 1. Since uv ∈ B, we have a + j ∈ Cφ (v ) = Cφ (u). Since φ (u) ̸ = φ (v ) = a + j, there is an edge uu1 colored with a + j. By Claim 3.2, uu1 ∈ A. Now we need to show a = φ (u). Otherwise, assume φ (u) = a + i, where i ≥ 1 and i ̸ = j. Thus a + i ∈ Cφ (u) = Cφ (v ). Since v is colored with a + j, there must be an edge vv1 colored with a + i. By Claim 3.2, we have φ (u1 ) = φ (v1 ) = a, a + j = φ (uu1 ) = φ (u) + 1 = a + i + 1, and a + i = φ (vv1 ) = φ (v ) + 1 = a + j + 1. Therefore a + i = a + i + 2. Since b ≥ 6, b − 1 ≥ 5. a + i ̸ = a + i + 2 (mod b − 1). This contradiction shows φ (u) = a and thus completes the proof of the claim. ■ For each edge uv ∈ B with φ (u) = a, φ (uv ) = t and φ (v ) = a + i for some 1 ≤ i ≤ b − 1, we recolor u with t and uv with φ (v ) + 2 = a + i + 2 (a + b = a + 1 and a + b + 1 = a + 2). Denote the new coloring ψ . The following two claims simply follow from the definitions of the coloring ψ , A and B. Claim 3.4. (1) For each edge uv , ψ (uv ) = φ (uv ) for each edge uv ̸ ∈ B. (2) For each vertex x ∈ V (G), ψ (x) = φ (x) unless x is incident with an edge xv ∈ B and φ (x) = a. In this case, ψ (x) = t ∈ {1, 2, . . . , a − 1}, ψ (xv ) = φ (v ) + 2, ψ (v ) = φ (v ) = a + j for some j ≥ 1. Moreover, t ̸∈ Cψ (v ) and thus Cψ (x) ̸ = Cψ (v ). Claim 3.5. In the coloring ψ , (1) uv ∈ B with ψ (u) < ψ (v ) if and only if ψ (u) ∈ {1, 2, . . . , a − 1}, ψ (v ) = a + j for some j ≥ 1 and ψ (uv ) = ψ (v ) + 2 = a + j + 2. (2) uv ∈ A with ψ (u) < ψ (v ) if and only if ψ (u) = a, ψ (v ) = a + j for some j ≥ 1 and ψ (uv ) = ψ (v ) + 1 = a + j + 1. (3) If uv ̸ ∈ A ∪ B with ψ (u) < ψ (v ), then ψ (uv ) = φ (uv ) ∈ {1, 2, . . . , a}, ψ (u) = φ (u) and ψ (v ) = φ (v ) = a + j for some j ≥ 1. By Claims 3.4 and 3.5, we have the following claim. Claim 3.6. ψ is a proper (a + b − 1)-total coloring of G. Claim 3.7. If uv is an edge in G with Cψ (u) = Cψ (v ), then uv ∈ A. Proof. Let uv be an edge with ψ (u) < ψ (v ) and Cψ (u) = Cψ (v ). Then ψ (v ) = φ (v ) = a + j for some j ≥ 1. By Claim 3.4, uv ̸ ∈ B. Suppose uv ̸ ∈ A. Then ψ (uv ) = φ (uv ) by the definitions of A, B and ψ, φ . We consider two cases depending on whether ψ (u) ≤ a or ψ (u) > a. If ψ (u) > a, we may assume that ψ (u) = a + i, then i < j. Since Cψ (u) = Cψ (v ), there must be an edge vv1 colored with a + i and an edge uu1 colored with a + j. By Claim 3.5, we have vv1 , uu1 ∈ A ∪ B. If vv1 ∈ A, then ψ (v1 ) = a, a + i = ψ (vv1 ) = ψ (v ) + 1 = a + j + 1, and if vv1 ∈ B, then ψ (v1 ) < a, a + i = ψ (vv1 ) = ψ (v ) + 2 = a + j + 2. If uu1 ∈ A,
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then ψ (u1 ) = a, a + j = ψ (uu1 ) = ψ (u) + 1 = a + i + 1, and if uu1 ∈ B, then ψ (u1 ) < a, a + j = ψ (uu1 ) = ψ (u) + 2 = a + i + 2. Therefore, a + i = a + i + 2 if vv1 , uu1 ∈ A, a + i = a + i + 3 if vv1 ∈ A, uu1 ∈ B or vv1 ∈ B, uu1 ∈ A, and a + i = a + i + 4 if vv1 , uu1 ∈ B. Since b ≥ 6, b−1 ≥ 5, then a+i ̸= a+i+2 (mod b−1), a+i ̸= a+i+3 (mod b−1), a+i ̸= a+i+4 (mod b−1), a contradiction. Now we assume ψ (u) ≤ a. Since v is colored with a + j and Cψ (u) = Cψ (v ), there is an edge uu1 colored with a + j. Then uu1 ∈ A ∪ B by Claim 3.5. If uu1 ∈ B, then ψ (u) < a, ψ (u1 ) ≥ a + 1, φ (u) = a and ψ (uu1 ) = ψ (u1 ) + 2. By Claim 3.3, there must be an edge uu2 ∈ A with ψ (uu2 ) = ψ (u1 ). Since ψ (uu2 ) ∈ Cψ (u) = Cψ (v ), there must be an edge vv1 colored with ψ (uu2 ). Then vv1 ∈ A ∪ B by Claim 3.5. If vv1 ∈ A, then ψ (vv1 ) = ψ (v ) + 1 = a + j + 1, and if vv1 ∈ B, then ψ (vv1 ) = ψ (v ) + 2 = a + j + 2. Noting that a + j = ψ (uu1 ) = ψ (u1 ) + 2 = ψ (uu2 ) + 2 = ψ (vv1 ) + 2, we have a + j = a + j + 3 if vv1 ∈ A, and a + j = a + j + 4 if vv1 ∈ B. Since b ≥ 6, b − 1 ≥ 5, then a + j ̸ = a + j + 3 (mod b − 1), a + j ̸ = a + j + 4 (mod b − 1), a contradiction. Therefore uu1 ∈ A. Then ψ (u) = a and φ (uu1 ) = a by Claim 3.5. Since a ∈ Cψ (u) = Cψ (v ), there must be an edge vv1 colored with a. Noting that uv ̸ ∈ B, there is an edge e adjacent to u or v such that e ∈ A ∪ B. Since ψ (e) ∈ Cψ (u) = Cψ (v ), there must be an edge uu2 colored with ψ (e), an edge vv2 colored with ψ (e), then uu2 , vv2 ∈ A ∪ B. Noting that uu1 ∈ A and A is a matching, we have uu2 ∈ B, then by Claim 3.5, ψ (u) < a, a contradiction. ■ Claim 3.8. ψ is a total (a + b − 1)-avd-coloring. Proof. If there is uv ∈ E(G) such that Cψ (u) = Cψ (v ), then uv ∈ A by Claim 3.7. By Claim 3.5, assume without loss of generality that ψ (u) = a, ψ (v ) = a + j for some j ≥ 1, then φ (uv ) = a and ψ (uv ) = ψ (v ) + 1 = a + j + 1. Since v is colored with a + j and Cψ (u) = Cψ (v ), there must be an edge uu1 colored with a + j. Then uu1 ∈ A ∪ B. Noting that uv ∈ A and A is a matching, we have uu1 ∈ B, then by Claim 3.5, ψ (u) < a, a contradiction. Therefore, we show that Cϕ (u) ̸ = Cϕ (v ) for any uv ∈ E(G), i.e., ϕ is a total-(a + b − 1)-avd-coloring and therefore completes the proof of Theorem 1.4. ■ Acknowledgments The first author’s research was partially supported by NSFC (No. 11601176) and NSF of Hubei Province (No. 2016CFB146). The second author’s research was partially supported by NSFC (No. 11671198). The third author’s research was partially supported by NSFC (Nos. 11571149 and 11571168). References [1] K. Appel, W. Haken, Every planar map is four colorable, Bull. Amer. Math. Soc. 82 (1976) 711–712. [2] X. Chen, On the adjacent vertex distinguishing total coloring numbers of graphs with ∆ = 3, Discrete Math. 308 (17) (2008) 4003–4007. [3] H. Grötzsch, Zur Theorie der diskreten Gebilde. VII. Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel. (German), Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg. Math.-Natur. Reih. 8 (1958/1959) 109–120. [4] D. Huang, W. Wang, C. Yan, A note on the adjacent vertex distinguishing total chromatic number of graphs, Discrete Math. 312 (24) (2012) 3544–3546. [5] J. Hulgan, Concise proofs for adjacent vertex-distinguishing total colorings, Discrete Math. 309 (2009) 2548–2550. [6] Z. Miao, R. Shi, X. Hu, R. Luo, Adjacent vertex distinguishing total colorings of 2-degenerate graphs, Discrete Math. 339 (2016) 2446–2449. [7] V. Vizing, On an estimate of the chromatic index of a p-graph, Diskret. Anal. 3 (1964) 25–30. [8] H. Wang, On the adjacent vertex distinguishing total chromatic number of the graphs with ∆ = 3, J. Comb. Optim. 14 (2007) 87–109. [9] W. Wang, D. Huang, The adjacent vertex distinguishing total coloring of planar graphs, J. Comb. Optim. 27 (2) (2014) 379–396. [10] W. Wang, Y. Wang, Adjacent vertex distinguishing total coloring of graphs with lower average degree, Taiwanese J. Math. 12 (4) (2008) 979–990. [11] W. Wang, P. Wang, On adjacent-vertex-distinguishing total coloring of K4 -minor-free graphs, Sci. China Ser. A 39 (12) (2009) 1462–1472 (in Chinese). [12] Y. Wang, W. Wang, Adjacent vertex distinguishing total colorings of outerplanar graphs, J. Comb. Optim. 19 (2010) 123–133. [13] Z. Zhang, X. Chen, J. Li, B. Yao, X. Lu, J. Wang, On adjacent-vertex-distinguishing total coloring of graphs, Sci. China Ser. A 48 (2005) 289–299.