Neighbor sum distinguishing total coloring of IC-planar graphs with short cycle restrictions

摘要

A graph is IC-planar if it admits a drawing in the plane with at most one crossing per edge, such that two pairs of crossing edges share no common end vertex. For a given graph G, a proper total coloring phi : V(G) boolean OR E(G) -> {1, 2,..., k} is neighbor sum distinguishing if f(phi)(u) not equal f(phi)(v) for each uv is an element of E(G), where f(phi)(v) = Sigma(uv is an element of E(G)) phi(uv) + phi(v), v is an element of V(G). The smallest integer k in such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by chi(Sigma)''(G). In this paper, by using the discharging method, we prove that chi(Sigma)''(G) <= max{Delta(G) + 3, 10} if G is a triangle free IC-planar graph and chi(Sigma)''(G) = max{Delta(G) + 3, 13} if G is an IC-planar graph without adjacent triangles, where Delta(G) is the maximum degree of G. (C) 2020 Elsevier B.V. All rights reserved.