Adjacent vertex distinguishing indices of planar graphs without 3-cycles

Abstract

Given a proper edge k-coloring ϕ and a vertex v∈V(G), let Cϕ(v) denote the set of colors used on edges incident to v with respect to ϕ. The adjacent vertex distinguishing index of G, denoted by χa′(G), is the least value of k such that G has a proper edge k-coloring which satisfies Cϕ(u)≠Cϕ(v) for any pair of adjacent vertices u and v. In this paper, we show that if G is a connected planar graph with maximum degree Δ≥12 and without 3-cycles, then Δ≤χa′(G)≤Δ+1, and χa′(G)=Δ+1 if and only if G contains two adjacent vertices of maximum degree. This extends a result in Edwards et al. (2006), which says that if G is a connected bipartite planar graph with Δ≥12 then χa′(G)≤Δ+1.