Neighbor Sum Distinguishing Total Choosability of 1-planar Graphs with Maximum Degree at Least 15

Abstract

Given a simple graph G = (V, E) and its (proper) total coloring ϕ with elements of the set {1, 2, ⋯, k}, let w_ϕ(v) denote the sum of the color of v and the colors of all edges incident with v. If for each edge uv ∈ E, w_ϕ(u) ≠ w_ϕ(v), we call ϕ a neighbor sum distinguishing total coloring of G. Let L = {L_x ∣ x ∈ V ⋃ E} be a set of lists of real numbers, each of size k. The neighbor sum distinguishing total choosability of G is the smallest k for which for any specified collection of such lists, there exists a neighbor sum distinguishing total coloring using colors from L_x for each x ∈ V ⋃ E, and we denote it by \(\text{ch}_{\sum}^{\prime\prime}(G)\). The known results of neighbor sum distinguishing total choosability are mainly about planar graphs. In this paper, we focus on 1-planar graphs. A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. We prove that \(\text{ch}_{\sum}^{\prime\prime}(G)\leq\Delta+4\) for any 1-planar graph G with Δ ≥ 15, where Δ is the maximum degree of G.