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India-Taiwan Conference on Discrete Mathematics, NTU, November 9–12, 2009 1 Adjacent vertex distinguishing edge-colorings of planar graphs with girth at least six Yuehua Bu1 yhbu@zjnu.cn Ko-Wei Lih2 makwlih@sinica.edu.tw Weifan Wang1 wwf@zjnu.cn 1 Department of Mathematics, Zhejiang Normal University, Zhejiang, Jinhua 321004, China 2 Institute of Mathematics, Academia Sinica, Nankang, Taipei 115, Taiwan Let G be a graph with vertex set V (G) and edge set E(G). A proper k-edge- coloring is a mapping φ : E(G) → {1, 2, . . . , k} such that φ(e) = φ(e ) for any two adjacent edges e and e . Let Cφ (v) = {φ(xv) | xv ∈ E(G)} denote the set of colors assigned to edges incident to the vertex v. A proper k-edge-coloring φ of G is adjacent vertex distinguishing, or a k-avd-coloring, if Cφ (u) = Cφ (v) whenever u and v are adjacent vertices. The adjacent vertex distinguishing chromatic index, denoted χa (G), is the smallest integer k such that G has a k-avd-coloring. Adjacent vertex distinguishing colorings are variously known as adjacent strong edge coloring [5] and 1-strong edge coloring [1]. Note that an isolated edge has no avd-coloring and a k-avd-coloring can be regarded as an m-avd-coloring for any m ≥ k. The chromatic index χ (G) of a graph G is the smallest integer k such that G has a proper k-edge-coloring. Evidently, χa (G) ≥ χ (G). Let ∆(G) denote the maximum degree of G. The well-known Vizing Theorem [4] asserts that ∆(G) ≤ χ (G) ≤ ∆(G) + 1 for every graph G. In contrast, there exist inﬁnitely many graphs G such that χa (G) > ∆(G) + 1. For instance, it is proved in [5] that, if n ≡ 0 (mod 3) and n = 5, then the cycle Cn satisﬁes χa (Cn ) = 4 = ∆(Cn ) + 2. However, χa (C5 ) = 5 = ∆(C5 ) + 3. Zhang, Liu, and Wang [5] completely determined the adjacent vertex distinguish- ing chromatic indices for paths, cycles, trees, complete graphs, and complete bipartite graphs. Based on these examples, they proposed the following conjecture. Conjecture 1. If G is a connected graph with at least 6 vertices, then χa (G) ≤ ∆(G) + 2. o Balister, Gy˝ri, Lehel, and Schelp [2] established the following three theorems. Theorem 1 If G is a graph without isolated edges and ∆(G) = 3, then χa (G) ≤ 5. Theorem 2. If G is a bipartite graph without isolated edges, then χa (G) ≤ ∆(G) + 2. Theorem 3. If G is a graph without isolated edges and the chromatic number of G 2 India-Taiwan Conference on Discrete Mathematics, NTU, November 9–12, 2009 is k, then χa (G) ≤ ∆(G) + O(log k). The following bound proved by Hatami [3] is better than Theorem 3 for graphs with extremely large chromatic numbers. Theorem 4. If G is a graph without isolated edges and ∆(G) > 1020 , then χa (G) ≤ ∆(G) + 300. In this paper, we prove Conjecture 1 for the following case. Theorem 5. If G is a plane graph without isolated edges whose girth g(G) ≥ 6, then χa (G) ≤ ∆(G) + 2. Our proof proceeds by reductio ad absurdum. Assume that G is a counterexample to the theorem whose |V (G)| + |E(G)| is the least possible. We are going to analyze the structure of G with a sequence of auxiliary lemmas. Then we will derive a contradiction using the discharging method. Lemma 1. No vertex of degree 2 is adjacent to a leaf. Lemma 2. If the degree dG (v) of v in G is at least 3, then there are at least three neighbors of v that are not leaves. A path x0 x1 · · · xk xk+1 of length k + 1 in G is called a k-chain if dG (x0 ) ≥ 3, dG (xk+1 ) ≥ 3, and dG (xi ) = 2 for all i = 1, 2, . . . , k. Lemma 3. There does not exist any k-chain if k ≥ 3. Lemma 4. There exists no edge xy with dG (x) = 2 and DG (y) = 3, where DG (y) denotes the number of neighbors of y that are not leaves. Lemma 5. There does not exist a vertex v with neighbors v1 , v2 , . . . , vk , k ≥ 4, such that dG (v1 ) = dG (v2 ) = 2, dG (v3 ) ≥ 2, dG (v4 ) ≥ 2, and dG (vi ) = 1 for all i = 5, 6, . . . , k. Lemma 6 There does not exist a vertex v with neighbors v1 , v2 , . . . , vk , k ≥ 5, such that dG (v1 ) = dG (v2 ) = dG (v3 ) = 2, dG (v4 ) ≥ 2, dG (v5 ) ≥ 2, and dG (vi ) = 1 for all i = 6, 7, . . . , k. Lemma 7. There does not exist a face f = [x1 x2 · · · x6 ] such that dG (xi ) = 2 for all xi except x1 and x4 . Let H be the graph obtained by removing all leaves of G. Then H is a connected proper subgraph of G. It follows from Lemmas 1 and 2 that, for every v ∈ V (H), dH (v) ≥ 2 and dH (v) = dG (v) if 2 ≤ dG (v) ≤ 3. Using v∈V (H) dH (v) = f ∈F (H) dH (f ) = 2|E(H)| and Euler’s formula |V (H)| − |E(H)| + |F (H)| = 2, we can derive the following identity. India-Taiwan Conference on Discrete Mathematics, NTU, November 9–12, 2009 3 (2dH (v) − 6) + (dH (f ) − 6) = −12. v∈V (H) f ∈F (H) We deﬁne a weight function w by w(v) = 2dH (v) − 6 for v ∈ V (H) and w(f ) = dH (f )−6 for f ∈ F (H). It follows from the above identity that the sum of all weights is equal to −12. We will design appropriate discharging rules and then redistribute weights accordingly. Once the discharging is ﬁnished, a new weight function w is produced. The sum of all weights is kept ﬁxed when the discharging is in progress. However, the outcome w (x) is nonnegative for all x ∈ V (H) ∪ F (H). This leads to the following obvious contradiction. 0≤ w (x) = w(x) = −12. x∈V (H)∪F (H) x∈V (H)∪F (H) There are two discharging rules. (R1) If v is a vertex of degree 2 incident to a face f , then f sends 1 to v for each occurrence of v in the boundary walk of f . A face f = [uvw · · ·] of H is called a light face belonging to v if dH (v) ≥ 4 and either dH (u) = 2 or dH (w) = 2. (R2) If dH (v) ≥ 4, then v sends 1 to each light face belonging to v. References [1] S. Akbari, H. Bidkhori, and N. Nosrati, r-strong edge colorings of graphs, Discrete Math. 306 (2006) 3005-3010. o [2] P. N. Balister, E. Gy˝ri, J. Lehel, and R. H. Schelp, Adjacent vertex distinguish- ing edge-colorings, SIAM J. Discrete Math. 21 (2007) 237-250. [3] H. Hatami, ∆ + 300 is a bound on the the adjacent vertex distinguishing edge chromatic number, J. Combin. Theory Ser. B 95 (2005) 246-256. [4] V. G. Vizing, On an estimate of the chromatic index of a p-graph, Diskret Analiz. 3 (1964) 25-30. [5] Z. Zhang, L. Liu, and J. Wang, Adjacent strong edge coloring of graphs, Appl. Math. Lett. 15 (2002) 623-626.