Adjacent Vertex Distinguishing Edge Coloring of Planar Graphs Without 4-Cycles

Abstract

The adjacent vertex distinguishing edge coloring of a graph G is a proper edge coloring of G such that the edge coloring set on any pair of adjacent vertices is distinct. The minimum number of colors required for an adjacent vertex distinguishing edge coloring of G is denoted by $$\chi _{a}'(G)$$. It is observed that $$\chi _a'(G)\ge \Delta (G)+1$$ when G contains two adjacent vertices of degree $$\Delta (G)$$. In this paper, we prove that if G is a planar graph without 4-cycles, then $$\chi _a'(G)\le \max \{9,\Delta (G)+1\}$$.