Neighbor sum distinguishing total colorings of planar graphs

A total [k]-coloring of a graph $$G$$ G is a mapping $$\phi : V (G) \cup E(G)\rightarrow [k]=\{1, 2,\ldots , k\}$$ ϕ : V ( G ) ∪ E ( G ) → [ k ] = { 1 , 2 , … , k } such that any two adjacent or incident elements in $$V (G) \cup E(G)$$ V ( G ) ∪ E ( G ) receive different colors. Let $$f(v)$$ f ( v ) denote the sum of the color of a vertex $$v$$ v and the colors of all incident edges of $$v$$ v . A total $$[k]$$ [ k ] -neighbor sum distinguishing-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 ) , $$f(u)\ne f(v)$$ f ( u ) ≠ f ( 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 conjectured $$\chi _{nsd}^{''}(G)\le \Delta (G)+3$$ χ n s d ′ ′ ( G ) ≤ Δ ( G ) + 3 for any simple graph with maximum degree $$\Delta (G)$$ Δ ( G ) . In this paper, we prove that this conjecture holds for any planar graph with maximum degree at least 13.