Adjacent vertex distinguishing total coloring of planar graphs with maximum degree 8

Abstract A (proper) total- k -coloring ϕ : V ( G ) ∪ E ( G ) → { 1 , 2 , … , k } is called adjacent vertex distinguishing if C ϕ ( u ) ≠ C ϕ ( v ) for each edge u v ∈ E ( G ) , where C ϕ ( u ) is the set of the color of u and the colors of all edges incident with u . We use χ a ′ ′ ( G ) to denote the smallest value k in such a coloring of G . Zhang et al. (2005) first introduced this coloring and conjectured that χ a ′ ′ ( G ) ≤ Δ ( G ) + 3 for every simple graph G . It is known that χ a ′ ′ ( G ) ≤ Δ ( G ) + 3 for every planar graph with Δ ( G ) ≥ 9 . In this paper, we succeed in proving the conjecture for every planar graph with Δ ( G ) ≥ 8 by using Alon’s Combinatorial Nullstellensatz and discharging method.