Acyclic edge coloring of planar graphs with Δ colors

Authors:
Dávid Hudák;František Kardoš;Borut Luar;Roman Soták;Riste Škrekovski
Affiliations:
Institute of Mathematics, Faculty of Science, Pavol Jozef Šafárik University, Košice, Slovakia;Institute of Mathematics, Faculty of Science, Pavol Jozef Šafárik University, Košice, Slovakia;Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia;Institute of Mathematics, Faculty of Science, Pavol Jozef Šafárik University, Košice, Slovakia;Department of Mathematics, Faculty of Mathematics and Physics, University of Ljubljana, Slovenia
Venue:
Discrete Applied Mathematics
Year:
2012

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Abstract

An acyclic edge coloring of a graph is a proper edge coloring without bichromatic cycles. In 1978, it was conjectured that @D(G)+2 colors suffice for an acyclic edge coloring of every graph G (Fiamcik, 1978 [8]). The conjecture has been verified for several classes of graphs, however, the best known upper bound for as special class as planar graphs are, is @D+12 (Basavaraju and Chandran, 2009 [3]). In this paper, we study simple planar graphs which need only @D(G) colors for an acyclic edge coloring. We show that a planar graph with girth g and maximum degree @D admits such acyclic edge coloring if g=12, or g=8 and @D=4, or g=7 and @D=5, or g=6 and @D=6, or g=5 and @D=10. Our results improve some previously known bounds.