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A note on acyclic edge coloring of complete bipartite graphs

Basavaraju, Manu and Chandran, Sunil L (2009) A note on acyclic edge coloring of complete bipartite graphs. In: Discrete Mathematics, 309 (13). pp. 4646-4648.

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An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic (2-colored) 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). Let Delta = Delta(G) denote the maximum degree of a vertex in a graph G. A complete bipartite graph with n vertices on each side is denoted by K-n,K-n. Alon, McDiarmid and Reed observed that a'(K-p-1,K-p-1) = p for every prime p. In this paper we prove that a'(K-p,K-p) <= p + 2 = Delta + 2 when p is prime. Basavaraju, Chandran and Kummini proved that a'(K-n,K-n) >= n + 2 = Delta + 2 when n is odd, which combined with our result implies that a'(K-p,K-p) = p + 2 = Delta + 2 when p is an odd prime. Moreover we show that if we remove any edge from K-p,K-p, the resulting graph is acyclically Delta + 1 = p + 1-edge-colorable. (C) 2009 Elsevier B.V. All rights reserved.

Item Type: Journal Article
Publication: Discrete Mathematics
Publisher: Elsevier Science BV
Additional Information: Copyright for this article belongs to Elsevier Science BV.
Keywords: Acyclic Edge Coloring; Acyclic Edge Chromatic Index; Matching; Complete Bipartite Graphs
Department/Centre: Division of Electrical Sciences > Computer Science & Automation
Date Deposited: 04 Dec 2009 05:31
Last Modified: 28 Feb 2019 08:39
URI: http://eprints.iisc.ac.in/id/eprint/21130

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