Neighbor sum distinguishing total coloring of IC-planar graphs

Abstract

Two distinct crossings are independent if the end-vertices of the crossed edge are mutually different. A graph is said to be plane graph with independent crossings or IC-planar, if it can be drawn in the plane such that every two crossings are independent. A proper total-k-coloring of G is called neighbor sum distinguishing if ∑c(u)≠∑c(v) for each edge uv∈E(G), where ∑c(v) is used to denote the sum of the color of a vertex v and the colors of edges incident with v. The least number k needed for such a coloring of G, denoted by χΣ′′(G), is the neighbor sum distinguishing total chromatic number. By using the famous Combinatorial Nullstellensatz, we prove that Δ(G)+2 colors guarantee such a total coloring for any IC-planar graphs without 2-vertex incident with crossed edge and with Δ(G)≥14.