Neighbor sum distinguishing total colorings of planar graphs

total [k]-coloring of a graph G is a mapping ϕ : V (G) ∪ E(G)→ [k]={1, 2,… , k} such that any two adjacent or incident elements in V (G) ∪ E(G) receive different colors. Let f(v) denote the sum of the color of a vertex v and the colors of all incident edges of v . A total [k] -neighbor sum distinguishing-coloring of G is a total [k] -coloring of G such that for each edge uv∈ E(G) , f(u) f(v) . By χ ^”_nsd(G) , we denote the smallest value k in such a coloring of G . Pilśniak and Woźniak conjectured χ _nsd^”(G)≤Δ (G)+3 for any simple graph with maximum degree Δ (G) . In this paper, we prove that this conjecture holds for any planar graph with maximum degree at least 13.