Neighbor sum distinguishing total coloring of graphs with bounded treewidth

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