Neighbor sum distinguishing total colorings of planar graphs

A total [k]-coloring of a graph $$G$$G is a mapping $$\phi : V (G) \cup E(G)\rightarrow [k]=\{1, 2,\ldots , k\}$$?: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 $$uv\in E(G)$$uv?E(G), $$f(u)\ne f(v)$$f(u)?f(v). By $$\chi ^{''}_{nsd}(G)$$?nsd??(G), we denote the smallest value $$k$$k in such a coloring of $$G$$G. Pilśniak and Wo?niak conjectured $$\chi _{nsd}^{''}(G)\le \Delta (G)+3$$?nsd??(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.