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# d-Regular Graphs of Acyclic Chromatic Index at Least d+2

Basavaraju, Manu and Chandran, Sunil L and Kummini, Manoj (2010) d-Regular Graphs of Acyclic Chromatic Index at Least d+2. In: Journal of Graph Theory, 63 (3). pp. 226-230.

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## Abstract

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, Suclakov and Zaks (and earlier by Fiamcik) that a'(G) &lt;= Delta+2, where Delta = Delta(G) denotes the maximum degree of the graph. Alon et al. also raised the question whether the complete graphs of even order are the only regular graphs which require Delta+2 colors to be acyclically edge colored. In this article, using a simple counting argument we observe not only that this is not true, but in fact all d-regular graphs with 2n vertices and d&gt;n, requires at least d+2 colors. We also show that a'(K-n,K-n) &gt;= n+2, when n is odd using a more non-trivial argument. (Here K-n,K-n denotes the complete bipartite graph with n vertices on each side.) This lower bound for Kn,n can be shown to be tight for some families of complete bipartite graphs and for small values of n. We also infer that for every d, n such that d &gt;= 5, n &gt;= 2d+3 and dn even, there exist d-regular graphs which require at least d+2-colors to be acyclically edge colored. (C) 2009 Wiley Periodicals, Inc. J Graph Theory 63: 226-230, 2010.

Item Type: Journal Article Copyright of this article belongs to John Wiley and Sons. acyclic edge coloring; acyclic edge chromatic index; matching; perfect 1-factorization; complete bipartite graphs Division of Electrical Sciences > Computer Science & Automation Id for Latest eprints 29 Mar 2010 11:44 28 Feb 2019 08:38 http://eprints.iisc.ac.in/id/eprint/26236 View Item