On r-acyclic edge colorings of planar graphs

  • Authors:

  • Xin Zhang;Guanghui Wang;Yong Yu;Jinbo Li;Guizhen Liu

  • Affiliations:

  • School of Mathematics, Shandong University, Jinan 250100, PR China and Department of Mathematics, Xidian University, Xi'an 710071, PR China;School of Mathematics, Shandong University, Jinan 250100, PR China;School of Mathematics, Shandong University, Jinan 250100, PR China;College of Sciences, China University of Mining and Technology, Xuzhou 221116, PR China;School of Mathematics, Shandong University, Jinan 250100, PR China

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2012

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Abstract

A proper edge coloring of G is r-acyclic if every cycle C contained in G is colored with at least min{|C|,r} colors. The r-acyclic chromatic index of a graph, denoted by a"r^'(G), is the minimum number of colors required to produce an r-acyclic edge coloring. In this paper, we study 4-acyclic edge colorings by proving that a"4^'(G)@?37@D(G) for every planar graph, a"4^'(G)@?max{2@D(G),3@D(G)-4} for every series-parallel graph and a"4^'(G)@?2@D(G) for every outerplanar graph. In addition, we prove that every planar graph with maximum degree at least r and girth at least 5r+1 has a"r^'(G)=@D(G) for every r=4.