Neighbor sum distinguishing total coloring of graphs with bounded treewidth

Abstract

A proper total k-coloring $\phi$ of a graph G is a mapping from $V(G)\cup E(G)$ to $\{1,2,\dots , k\}$ such that no adjacent or incident elements in $V(G)\cup E(G)$ receive the same color. Let $m_{\phi }(v)$ denote the sum of the colors on the edges incident with the vertex v and the color on v. A proper total k-coloring of G is called neighbor sum distinguishing if $m_{\phi }(u)\not =m_{\phi }(v)$ for each edge $uv\in E(G).$ Let $\chi _{\Sigma }^t(G)$ be the neighbor sum distinguishing total chromatic number of a graph G. Pilśniak and Woźniak conjectured that for any graph G, $\chi _{\Sigma }^t(G)\le \Delta (G)+3$. In this paper, we show that if G is a graph with treewidth $\ell \ge 3$ and $\Delta (G)\ge 2\ell +3$, then $\chi _{\Sigma }^t(G)\le \Delta (G)+\ell -1$. This upper bound confirms the conjecture for graphs with treewidth 3 and 4. Furthermore, when $\ell =3$ and $\Delta \ge 9$, we show that $\Delta (G) + 1\le \chi _{\Sigma }^t(G)\le \Delta (G)+2$ and characterize graphs with equalities.