Neighbor sum distinguishing total coloring of 2-degenerate graphs

Abstract

A proper k-total coloring of a graph G is a mapping from $V(G)\cup E(G)$ to $\{1,2,\ldots ,k\}$ such that no two adjacent or incident elements in $V(G)\cup E(G)$ receive the same color. Let f(v) denote the sum of the colors on the edges incident with v and the color on vertex v. A proper k-total coloring of G is called neighbor sum distinguishing if $f(u)\ne f(v)$ for each edge $uv\in E(G)$. Let $\chi ''_{\Sigma }(G)$ denote the smallest integer k in such a coloring of G. Pilśniak and Woźniak conjectured that for any graph G, $\chi ''_{\Sigma }(G)\le \Delta (G)+3$. In this paper, we show that if G is a 2-degenerate graph, then $\chi ''_{\Sigma }(G)\le \Delta (G)+3$; Moreover, if $\Delta (G)\ge 5$ then $\chi ''_{\Sigma }(G)\le \Delta (G)+2$.