Neighbor sum distinguishing total coloring of planar graphs without 5-cycles.

Let G be a graph, a proper total coloring ϕ:V(G)∪E(G)→{1,2,…,k} is called neighbor sum distinguishing if f(u)≠f(v) for each edge uv∈E(G), where f(v)=∑uv∈E(G)ϕ(uv)+ϕ(v), v∈V(G). We use χΣ″(G) to denote the smallest number k in such a coloring of G. Pilśniak and Woźniak have already conjectured that χΣ″(G)≤Δ(G)+3 for any simple graph with maximum degree Δ(G). In this paper, we prove that for any planar graph G without 5-cycles, χΣ″(G)≤max⁡{Δ(G)+3,10}.