Neighbor sum distinguishing total coloring of IC-planar graphs with short cycle restrictions

Abstract

A graph is IC-planar if it admits a drawing in the plane with at most one crossing per edge, such that two pairs of crossing edges share no common end vertex. For a given graph G, a proper total coloring ϕ : V(G)∪E(G)→{1,2,…,k} is neighbor sum distinguishing if fϕ(u)≠fϕ(v) for each uv∈E(G), where fϕ(v)=∑uv∈E(G)ϕ(uv)+ϕ(v), v∈V(G). The smallest integer k in such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by χΣ′′(G). In this paper, by using the discharging method, we prove that χΣ′′(G)≤max{Δ(G)+3,10} if G is a triangle free IC-planar graph and χΣ′′(G)≤max{Δ(G)+3,13} if G is an IC-planar graph without adjacent triangles, where Δ(G) is the maximum degree of G.