Adjacent Vertex Distinguishing I-Total Coloring of Graphs with Bounded Maximum Average Degree

A k-I-total coloring φ of a graph G is a mapping from the elements in V(G)∪E(G) to k colors such that no two adjacent vertices as well as edges receive same color in G, and moreover, if Cφ(u)≠Cφ(v) for any uv∈E(G), where Cφ(u) denotes the set of colors assigned to u and its incident edges under φ, then φ is termed k-adjacent vertex distinguishing I-total coloring of G, and the smallest positive integer k is called adjacent vertex distinguishing I-total chromatic number, denoted by χati(G) for short. Let Δ(G) and mad(G) be the maximum degree and the maximum average degree of G, respectively. In this paper, we employ the discharging method to prove that χati(G)≤max{7,Δ(G)+1} for any graph G with mad(G)<4.