Neighbor sum distinguishing total choosability of planar graphs without 4-cycles

Abstract

Let G=(V,E) be a graph and ϕ be a total k-coloring of G by using the color set {1,…,k}. Let ∑ϕ(u) denote the sum of the color of the vertex u and the colors of all incident edges of u. A k-neighbor sum distinguishing total coloring of G is a total k-coloring of G such that for each edge uv∈E(G), ∑ϕ(u)≠∑ϕ(v). By χΣ″(G), we denote the smallest value k in such a coloring of G. Pilśniak and Woźniak first introduced this coloring and conjectured that χΣ″(G)≤Δ(G)+3 for any simple graph G. Let Lz(z∈V∪E) be a set of lists of integer numbers, each of size k. The smallest k for which for any specified collection of such lists, there exists a neighbor sum distinguishing total coloring using colors from Lz for each z∈V∪E is called the neighbor sum distinguishing total choosability of G, and denoted by chΣ″(G). In this paper, we prove that chΣ″(G)≤Δ(G)+3 for planar graphs without 4-cycles with Δ(G)≥7. This implies that Pilśniak and Woźniak’ conjecture is true for planar graphs without 4-cycles.