On rr-acyclic edge colorings of planar graphs

A proper edge coloring of GG is rr-acyclic if every cycle CC contained in GG is colored with at least min{|C|,r}min{|C|,r} colors. The rr-acyclic chromatic index of a graph, denoted by ar′(G), is the minimum number of colors required to produce an rr-acyclic edge coloring. In this paper, we study 4-acyclic edge colorings by proving that a4′(G)≤37Δ(G) for every planar graph, a4′(G)≤max{2Δ(G),3Δ(G)−4} for every series–parallel graph and a4′(G)≤2Δ(G) for every outerplanar graph. In addition, we prove that every planar graph with maximum degree at least rr and girth at least 5r+15r+1 has ar′(G)=Δ(G) for every r≥4r≥4.