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Acyclic edge colorings of graphs

✍ Scribed by Noga Alon; Benny Sudakov; Ayal Zaks

Publisher: John Wiley and Sons
Year: 2001
Tongue: English
Weight: 102 KB
Volume: 37
Category: Article
ISSN: 0364-9024
DOI: 10.1002/jgt.1010

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✦ Synopsis

Abstract

A proper coloring of the edges of a graph G is called acyclic if there is no 2‐colored cycle in G. The acyclic edge chromatic number of G, denoted by a′(G), is the least number of colors in an acyclic edge coloring of G. For certain graphs G, a′(G) ≥ Δ(G) + 2 where Δ(G) is the maximum degree in G. It is known that a′(G) ≤ 16 Δ(G) for any graph G. We prove that there exists a constant c such that a′(G) ≤ Δ(G) + 2 for any graph G whose girth is at least __c__Δ(G) log Δ(G), and conjecture that this upper bound for a′(G) holds for all graphs G. We also show that a′(G) ≤ Δ + 2 for almost all Δ‐regular graphs. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 157–167, 2001

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