The adjacent vertex distinguishing total chromatic numbers of planar graphs with $$\Delta =10$$ Δ = 10

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 $$uv\in E(G)$$ u v ∈ E ( G ) , $$C(u)\ne C(v)$$ C ( u ) ≠ C ( v ) . We denote the smallest value k in such a coloring of G by $$\chi ''_{a}(G)$$ χ a ′ ′ ( G ) . It is known that $$\chi _{a}''(G)\le \Delta (G)+3$$ χ a ′ ′ ( 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 $$\chi _{a}''(G)\le \Delta (G)+3$$ χ a ′ ′ ( G ) ≤ Δ ( G ) + 3 . Our approach is based on Combinatorial Nullstellensatz and the discharging method.