Neighbor sum distinguishing total coloring of 2-degenerate graphs

A proper k-total coloring of a graph G is a mapping from $$V(G)\cup E(G)$$ V ( G ) ∪ E ( G ) to $$\{1,2,\ldots ,k\}$$ { 1 , 2 , … , k } such that no two adjacent or incident elements in $$V(G)\cup E(G)$$ V ( G ) ∪ 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 $$f(u)\ne f(v)$$ f ( u ) ≠ f ( v ) for each edge $$uv\in E(G)$$ u v ∈ E ( G ) . Let $$\chi ''_{\Sigma }(G)$$ χ Σ ′ ′ ( G ) denote the smallest integer k in such a coloring of G. Pilśniak and Woźniak conjectured that for any graph G, $$\chi ''_{\Sigma }(G)\le \Delta (G)+3$$ χ Σ ′ ′ ( G ) ≤ Δ ( G ) + 3 . In this paper, we show that if G is a 2-degenerate graph, then $$\chi ''_{\Sigma }(G)\le \Delta (G)+3$$ χ Σ ′ ′ ( G ) ≤ Δ ( G ) + 3 ; Moreover, if $$\Delta (G)\ge 5$$ Δ ( G ) ≥ 5 then $$\chi ''_{\Sigma }(G)\le \Delta (G)+2$$ χ Σ ′ ′ ( G ) ≤ Δ ( G ) + 2 .