A proper total coloring distinguishing adjacent vertices by sums of planar graphs without intersecting triangles

Let $$G=(V,E)$$ G = ( V , E ) be a graph and $$\phi $$ ϕ be a total $$k$$ k -coloring of $$G$$ G using the color set $$\{1,\ldots , k\}$$ { 1 , … , k } . Let $$\sum _\phi (u)$$ ∑ ϕ ( u ) denote the sum of the color of the vertex $$u$$ u and the colors of all incident edges of $$u$$ u . A $$k$$ k -neighbor sum distinguishing total coloring of $$G$$ G is a total $$k$$ k -coloring of $$G$$ G such that for each edge $$uv\in E(G)$$ u v ∈ E ( G ) , $$\sum _\phi (u)\ne \sum _\phi (v)$$ ∑ ϕ ( u ) ≠ ∑ ϕ ( v ) . By $$\chi ^{''}_{nsd}(G)$$ χ n s d ′ ′ ( G ) , we denote the smallest value $$k$$ k in such a coloring of $$G$$ G . Pilśniak and Woźniak first introduced this coloring and conjectured that $$\chi _{nsd}^{''}(G)\le \Delta (G)+3$$ χ n s d ′ ′ ( G ) ≤ Δ ( G ) + 3 for any simple graph $$G$$ G . In this paper, we prove that the conjecture holds for planar graphs without intersecting triangles with $$\Delta (G)\ge 7$$ Δ ( G ) ≥ 7 . Moreover, we also show that $$\chi _{nsd}^{''}(G)\le \Delta (G)+2$$ χ n s d ′ ′ ( G ) ≤ Δ ( G ) + 2 for planar graphs without intersecting triangles with $$\Delta (G) \ge 9$$ Δ ( G ) ≥ 9 . Our approach is based on the Combinatorial Nullstellensatz and the discharging method.