Neighbor sum distinguishing total choosability of planar graphs without adjacent special 5-cycles

Abstract

A total k-coloring of a graph G is a mapping ϕ: V(G)∪E(G)→{1,2,…,k} such that no two adjacent or incident elements in V(G)∪E(G) receive the same color. In a total k-coloring of G, let w(v) denote the total sum of colors of the edges incident with v and the color of v. If for each edge uv∈E(G), w(u)≠w(v), then we call such a total k-coloring neighbor sum distinguishing. Let χ∑′′(G) denote the smallest number k in such a coloring of G. Pilśniak and Woźniak posed the conjecture that χ∑′′(G)≤Δ(G)+3 for any simple graph G with maximum degree Δ(G). In this paper, we focus on the list version of neighbor sum distinguishing total coloring. Let Lz (z∈V(G)∪E(G)) 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(G)∪E(G) is called the neighbor sum distinguishing total choosability of G, and denoted by ch∑′′(G). We prove that ch∑′′(G)≤Δ(G)+3 for planar graphs without adjacent special 5-cycles with Δ(G)≥8.