Adjacent vertex distinguishing edge colorings of planar graphs with girth at least five

Abstract

An adjacent vertex distinguishing edge coloring of a graph $$G$$ is a proper edge coloring of $$G$$ such that any pair of adjacent vertices admit different sets of colors. The minimum number of colors required for such a coloring of $$G$$ is denoted by $$\chi ^{\prime }_{a}(G)$$ . In this paper, we prove that if $$G$$ is a planar graph with girth at least 5 and $$G$$ is not a 5-cycle, then $$\chi ^{\prime }_{a}(G)\le \Delta +2$$ , where $$\Delta $$ is the maximum degree of $$G$$ . This confirms partially a conjecture in Zhang et al. (Appl Math Lett 15:623---626, 2002).