Adjacent vertex distinguishing total colorings of 2-degenerate graphs

Abstract

Let ϕ be a proper total coloring of G. We use Cϕ(v)={ϕ(v)}∪{ϕ(uv)∣uv∈E(G)} to denote the set of colors assigned to a vertex v and those edges incident with v. An adjacent vertex distinguishing total coloring of a graph G is a proper total coloring of G such that Cϕ(u)≠Cϕ(v) for any uv∈E(G). The minimum number of colors required for an adjacent vertex distinguishing total coloring of G is denoted by χa″(G). In this paper we show that if G is a 2-degenerate graph, then χa″(G)≤max{Δ(G)+2,6}. Moreover, we also show that when Δ≥5, χa″(G)=Δ(G)+2 if and only if G contains two adjacent vertices of maximum degree. Our results imply the results on outerplanar graphs (Wang and Wang, 2010), K4-minor free graphs (Wang and Wang, 2009) and graphs with maximum average degree less than 3 (Wang and Wang, 2008).