A new upper bound on the acyclic chromatic indices of planar graphs

Authors:
Weifan Wang;Qiaojun Shu;Yiqiao Wang
Affiliations:
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China;Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China;School of Management, Beijing University of Chinese Medicine, Beijing 100029, China
Venue:
European Journal of Combinatorics
Year:
2013

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Abstract

An acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index a^'(G) of G is the smallest integer k such that G has an acyclic edge coloring using k colors. It was conjectured that a^'(G)@?@D+2 for any simple graph G with maximum degree @D. In this paper, we prove that if G is a planar graph, then a^'(G)@?@D+7. This improves a result by Basavaraju et al. [M. Basavaraju, L.S. Chandran, N. Cohen, F. Havet, T. Muller, Acyclic edge-coloring of planar graphs, SIAM J. Discrete Math. 25 (2011) 463-478], which says that every planar graph G satisfies a^'(G)@?@D+12.