Adjacent vertex distinguishing edge-colorings of graphs with smaller maximum average degree

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 are incident to distinct sets of colors. The minimum number of colors required for an adjacent vertex distinguishing edge-coloring of G is denoted by χ′ _a (G). Let $\mathop{\mathrm{mad}}(G)$ and Δ denote the maximum average degree and the maximum degree of a graph G, respectively.

In this paper, we prove the following results: (1) If $\mathop{\mathrm{mad}}(G)<3$ and Δ≥3, then χ′ _a (G)≤Δ+2. (2) If $\mathop{\mathrm{mad}}(G)<\frac{5}{2}$ and Δ≥4, or $\mathop{\mathrm{mad}}(G)<\frac{7}{3}$ and Δ=3, then χ′ _a (G)≤Δ+1. (3) If $\mathop{\mathrm{mad}}(G)<\frac{5}{2}$ and Δ≥5, then χ′ _a (G)=Δ+1 if and only if G contains adjacent vertices of maximum degree.