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Acyclic edge coloring of triangle-free planar graphs

✍ Scribed by Manu Basavaraju; L. Sunil Chandran

Publisher
John Wiley and Sons
Year
2011
Tongue
English
Weight
233 KB
Volume
71
Category
Article
ISSN
0364-9024

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✦ Synopsis

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). It was conjectured by Alon, Sudakov and Zaks (and much earlier by Fiamcik) that a′(G) ⩽ Δ + 2, where Δ = Δ(G) denotes the maximum degree of the graph. If every induced subgraph H of G satisfies the condition |E(H)| ⩽ 2|V(H)|−1, we say that the graph G satisfies Property A. In this article, we prove that if G satisfies Property A, then a′(G) ⩽ Δ + 3. Triangle‐free planar graphs satisfy Property A. We infer that a′(G) ⩽ Δ + 3, if G is a triangle‐free planar graph. Another class of graph which satisfies Property A is 2‐fold graphs (union of two forests). © 2011 Wiley Periodicals, Inc. J Graph Theory

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