Adjacent Vertex Distinguishing Total Coloring of Graphs with Lower Average Degree

An adjacent vertex distinguishing total coloring of a graph G is a proper total 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 total coloring of G is denoted by χ′′ a(G). Let mad(G) and ∆(G) denote the maximum average degree and the maximum degree of a graph G, respectively. In this paper, we prove the following results: (1) If G is a graph with mad(G) < 3 and ∆(G) ≥ 5, then ∆(G) + 1 ≤ χ′′ a(G) ≤ ∆(G) + 2, and χ′′ a(G) = ∆(G)+2 if and only ifG contains two adjacent vertices of maximum degree; (2) If G is a graph with mad(G) < 3 and ∆(G) ≤ 4, then χ′′ a(G) ≤ 6; (3) If G is a graph with mad(G) < 83 and ∆(G) ≤ 3, then χ′′ a(G) ≤ 5.