On r-acyclic edge colorings of planar graphs

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)@[email protected](G) for every planar graph, a"4^'(G)@?max{[email protected](G),[email protected](G)-4} for every series-parallel graph and a"4^'(G)@[email protected](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)[email protected](G) for every r>=4.