Neighbor sum distinguishing total coloring of planar graphs without 4-cycles

Let $$G=(V,E)$$ G = ( V , E ) be a graph and $$\phi : V\cup E\rightarrow \{1,2,\ldots ,k\}$$ ϕ : V ∪ E → { 1 , 2 , … , k } be a proper total coloring of G. Let f(v) denote the sum of the color on a vertex v and the colors on all the edges incident with v. The coloring $$\phi $$ ϕ is neighbor sum distinguishing if $$f(u)\ne f(v)$$ f ( u ) ≠ f ( v ) for each edge $$uv\in E(G)$$ u v ∈ E ( G ) . The smallest integer k in such a coloring of G is the neighbor sum distinguishing total chromatic number of G, denoted by $$\chi _{\Sigma }''(G)$$ χ Σ ′ ′ ( G ) . Pilśniak and Woźniak conjectured that $$\chi _{\Sigma }''(G)\le \Delta (G)+3$$ χ Σ ′ ′ ( G ) ≤ Δ ( G ) + 3 for any simple graph. By using the famous Combinatorial Nullstellensatz, we prove that $$\chi _{\Sigma }''(G)\le \max \{\Delta (G)+2, 10\}$$ χ Σ ′ ′ ( G ) ≤ max { Δ ( G ) + 2 , 10 } for planar graph G without 4-cycles. The bound $$\Delta (G)+2$$ Δ ( G ) + 2 is sharp if $$\Delta (G)\ge 8$$ Δ ( G ) ≥ 8 .