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

A total k-coloring of a graph G is a mapping phi: V(G) boolean OR E(G) -> {1, 2, ..., k} such that no two adjacent or incident elements in V(G) boolean OR 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 is an element of E(G), w(u) not equal w(v), then we call such a total k-coloring neighbor sum distinguishing. Let chi(Sigma)''(G) denote the smallest number k in such a coloring of G. Pilsniak and Wozniak posed the conjecture that chi(Sigma)''(G) <= Delta(G) + 3 for any simple graph G with maximum degree Delta(G). In this paper, we focus on the list version of neighbor sum distinguishing total coloring. Let L-z (z is an element of V(G) boolean OR 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 L-z for each z is an element of V(G) boolean OR E(G) is called the neighbor sum distinguishing total choosability of G, and denoted by ch(Sigma)''(G). We prove that ch(Sigma)''(G) <= Delta(G) + 3 for planar graphs without adjacent special 5-cycles with Delta(G) >= 8. (C) 2019 Elsevier B.V. All rights reserved.