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Strong Edge-Coloring for Planar Graphs with Large Girth Lily

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Strong Edge-Coloring for Planar Graphs with Large Girth Lily STRONG EDGE-COLORING FOR PLANAR GRAPHS WITH LARGE GIRTH LILY CHEN1, KECAI DENG1, GEXIN YU2;3, AND XIANGQIAN ZHOU1;4 1 School of Mathematical Sciences, Huaqiao University, China 2 School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei, China. 3 Department of Mathematics, College of William and Mary, Williamsburg, VA, USA. 4Department of Mathematics and Statistics, Wright State University, Dayton, Ohio, 45435 Abstract. A strong edge-coloring of a graph G = (V; E) is a partition of its edge set E into induced matchings. Let G be a planar graph with girth k ≥ 26 and maximum degree ∆. We show that either G is isomorphic to a subgraph of a very special ∆-regular graph 12(∆−2) with girth k, or G has a strong edge-coloring using at most 2∆ + d k e colors. 1. introduction Graphs in this article are assumed to be simple and undirected. Let G be a graph. A proper edge-coloring of G is an assignment of colors to the edges such that no two adjacent edges receive the same color. Clearly, every coloring class is a matching of G. However, these matchings may not be induced. A strong edge-coloring of a graph G, first introduced by Fouquet and Jolivet [6], is a proper edge-coloring such that every two edges joined by another edge are colored differently. In a strong edge-coloring, every color class is an induced matching. The minimum number of colors required in a strong edge-coloring of G is called 0 the strong chromatic index and is denoted by χs(G). Let e and e0 be two edges of G. We say that e sees e0 if e and e0 are adjacent or share a common adjacent edge. So, equivalently, a strong edge-coloring is an assignment of colors to all edges such that every two edges that can see each other receive distinct colors. Let ∆ be the maximum degree of G, and for u 2 V (G), let dG(u) denote the degree of u in G. In this paper, we study strong edge-coloring of planar graphs. Faudree et al [5] conjectured that every subcubic planar graph has a strong edge-coloring using at most 9 colors. This conjecture was recently confirmed by Kostochka et al [8]. For an arbitrary planar graph G, E-mail address: [email protected], [email protected]. Date: November 8, 2017. 1991 Mathematics Subject Classification. 05C15. Key words and phrases. planar graph, strong edge-coloring, girth. Both the first and the second author are supported by NSFC (No.11501223 and No.11701195), and Science Foundation of the Fujian Province, China (No.2016J05009); the first author is also supported by Scientific Research Funds of Huaqiao University (No.14BS311); and the second author is also supported by Research Funds of Huaqiao University (No.16BS808) and NSFC (No. 11471273). The third author is supported in part by the NSA grant: H98230-16-1-0316 and Natural Science Foundation of China (11728102). The fourth author is partially supported by the Minjiang Scholar Program hosted by Huaqiao University, Quanzhou, Fujian, China. 1 0 0 0 Faudree et al [5] proved that χs(G) ≤ 4χ (G), where χ (G) is the chromatic index of G. Now 0 it follows from Vizing's Theorem [10] that χs(G) ≤ 4∆ + 4. Faudree et al [5] also showed 0 that there exists a planar graph G with χs(G) = 4∆ − 4 for every integer value of ∆ ≥ 2. 0 Better bounds for χs(G) were obtained when G is required to have a large girth. The next result of Hud´aket al [7] is a good example of such results. Theorem 1.1. (Hud´aket al 2014) 0 1. If G is a planar graph with girth at least 7, then χs(G) ≤ 3∆(G). 0 2. If G is a planar graph with girth at least 6, then χs(G) ≤ 3∆(G) + 5. For planar graph with girth 6, the bound given in Theorem 1.1 was improved to 3∆ + 1 by Bensmail et al [1], and independently by Ruksasakchai and Wang [9], in 2014. 0 Note that, when a graph G has two adjacent degree-∆ vertices, then χs(G) ≥ 2∆ − 1. Let G be a planar graph with ∆ ≥ 3 and girth g. Borodin and Ivanova [2] proved in 2013 that ∆ 0 if g ≥ 40b 2 c + 1, then χs(G) ≤ 2∆ − 1. The bound on the girth of G was later improved to 10∆ + 46 by Chang et al [4]. The best bound on the girth was established by Wang and Zhao [11] in the following result. Theorem 1.2. (Wang and Zhao 2015) If G is a planar graph with ∆ ≥ 4 and girth at least 0 10∆ − 4, then χs(G) ≤ 2∆ − 1. Chang and Duh [3] introduced another parameter σ on a graph G defined as follows. σ(G) = max fdG(x) + dG(y) − 1g: xy2E(G) 0 It is clear that for any graph G, χs(G) ≥ σ(G). Chang and Duh [3] proved the following theorem. Theorem 1.3. (Chang and Duh 2015) Let G be a planar graph with σ(G) ≥ 5 and σ(G) ≥ 0 ∆(G) + 2. If the girth of G is at least 5σ(G) + 16, then χs(G) = σ(G). Note that, the bounds on the girth given in Theorem 1.2 and 1.3 are both larger than a linear function of ∆, and so they become really big for graphs with large ∆ values. It would be nice if one can improve the lower bound on the girth down to a constant instead of a linear function of ∆. However, Hud´aket al [7] showed that there exists a planar graph G 0 ∆ with girth k and with χs(G) > 2∆ by constructing the following examples: Let Gk be the graph obtained from a cycle Ck of length k by adding ∆ − 2 pendent edges at each vertex ˘ of Ck. Hud´ak,Lu˘zar,Sot´ak,and Skrekovski [7] proved the following result. Proposition 1.4. (Hud´akat al 2014) For every odd integer k ≥ 3 and every integer ∆ ≥ 3, the following inequality holds. 2k(∆ − 1) 2k(∆ − 2) ≤ χ0 (G∆) ≤ + 5: k − 1 s k k − 1 Motivated by these constructions, Hud´aket al [7] made the following conjecture. Conjecture 1.5. (Hud´akat al 2014) There exists a constant c such that for every planar graph G of girth k ≥ 5, 2(∆ − 1) χ0 (G) ≤ 2∆ + + c: s k − 1 2 In the paper, we prove the following theorem, which is a weaker result than Conjecture 1.5. Theorem 1.6. Let G be a planar graph with ∆(G) = ∆ ≥ 4 and girth g = k ≥ 26. Then ∆ either G is isomorphic to a subgraph of Gk , or 12(∆ − 2) χ0 (G) ≤ 2∆ + : s k We would like to point out that, with the exception of some small values of ∆, the upper 0 ∆ bound on χs(Gk ) given by Hud´aket al [7] in Proposition 1.4 is better than our bound given ∆ in Theorem 1.6. That is the reason we keep subgraphs of Gk as one of the outcomes in Theorem 1.6. The paper is organized as follows: we present the proof for Theorem 1.6 in Section 2 and then in Section 3, we talk about some possible extensions of our result. 2. proof of theorem 1.6 First we will introduce some notions to be used in our proof. A k-vertex is a vertex with degree exactly k. A vertex is called a k+-vertex (resp. k−-vertex) if it has degree at least k (resp. at most k). The notions of k-neighbor, k+-neighbor, and k−-neighbor of a vertex v are defined in a similar manner. A vertex v in a graph G is called a poor vertex if dG(v) ≥ 2 and v has exactly dG(v) − 2 1-neighbors. A vertex v is called a rich vertex if dG(v) ≥ 3 and v has at least three 2+-neighbors. To prove Theorem 1.6, we choose G to be a minimal counterexample, that is, G is a planar graph with girth g = k ≥ 26 and ∆(G) = ∆ such that 1) G does not have a strong 12(∆−2) edge-coloring using at most 2∆ + d k e colors; and 2) jV (G)j is as small as possible. We will first prove some structural results of G. 2.0.1. A 1-vertex must be adjacent to a 4+-vertex. Proof. Suppose u is a 1-vertex and whose neighbor v is a 3−-vertex. Then the edge uv sees at most 2∆ edges. Since we have more than 2∆ + 1 colors, a strong edge-coloring of Gnu can be easily extended to a strong edge-coloring of G, a contradiction. It follows from 2.0.1 that every 2-vertex is a poor vertex. 2.0.2. Every 2+-vertex of G has at least two 2+-neighbors. Proof. Let v be a 2+-vertex of G. If v has no 2+-neighbor, then since G is connected, G is isomorphic to a star, which has a strong edge coloring using ∆ colors. Next assume that v + has exactly one 2 -neighbor. Since dG(v) ≥ 2, v has at least one 1-neighbor. Let u be a 1-neighbor of v. Since G is a minimal counterexample, Gnu has a strong edge-coloring using 12(∆−2) at most 2∆ + d k e colors. The edge uv sees at most 2∆ − 2 edges in G, so a coloring of Gnu can be easily extended to a coloring of G, a contradiction.
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