Neighbor sum distinguishing total colorings of planar graphs with maximum degree Δ

A (proper) total [k]-coloring of a graph G is a mapping ϕ:V(G)∪E(G)→[k]={1,2,…,k} such that any two adjacent elements in V(G)∪E(G) receive different colors. Let f(v) denote the sum of the color of a vertex v and the colors of all incident edges of v. A total [k]-neighbor sum distinguishing-coloring of G is a total [k]-coloring of G such that for each edge uv∈E(G), f(u)≠f(v). By χnsd″(G), we denote the smallest value k in such a coloring of G. In this paper, we show that if G is a planar graph with Δ(G)≥14, then χnsd″(G)≤Δ(G)+2.