Neighbor sum distinguishing total colorings of IC-planar graphs with maximum degree 13

Abstract

A graph is IC-planar if it admits a drawing on 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 $$\phi $$ : $$V(G)~\cup ~E(G)\rightarrow \{1,2,\ldots ,k\}$$ is called neighbor sum distinguishing if $$f_{\phi }(u)\ne f_{\phi }(v)$$ for each $$uv\in E(G)$$, where $$f_{\phi }(u)$$ is the sum of the color of u and the colors of the edges incident with u. The smallest integer k in such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by $$\chi ''_{\Sigma }(G)$$. Pilśniak and Woźniak conjectured $$\chi _{\Sigma }''(G)\le \Delta (G)+3$$ for any simple graph with maximum degree $$\Delta (G)$$. This conjecture was confirmed for IC-planar graph with maximum degree at least 14. In this paper, by using the discharging method, we prove that this conjecture holds for any IC-planar graph G with $$\Delta (G)=13$$.