LAUSR

A (proper) total-k-coloring of a graph G is a mapping $$\phi : V (G) \cup E(G)\mapsto \{1, 2, \ldots , k\}$$ϕ:V(G)∪E(G)↦{1,2,…,k} such that any two adjacent elements in $$V (G) \cup E(G)$$V(G)∪E(G) receive different colors. Let C(v) denote the set of the color of a vertex v and the colors of all incident edges of v. A total-k-adjacent vertex distinguishing-coloring of G is a total-k-coloring of G such that for each edge $$uv\in E(G)$$uv∈E(G), $$C(u)\ne C(v)$$C(u)≠C(v). We denote the smallest value k in such a coloring of G by $$\chi ''_{a}(G)$$χa′′(G). It is known that $$\chi _{a}''(G)\le \Delta (G)+3$$χa′′(G)≤Δ(G)+3 for any planar graph with $$\Delta (G)\ge 11$$Δ(G)≥11. In this paper, we show that if G is a planar graph with $$\Delta (G)\ge 10$$Δ(G)≥10, then $$\chi _{a}''(G)\le \Delta (G)+3$$χa′′(G)≤Δ(G)+3. Our approach is based on Combinatorial Nullstellensatz and the discharging method.