Planar graphs with $$\Delta =9$$Δ=9 are neighbor-distinguishing totally 12-colorable

Abstract

The neighbor-distinguishing total coloring of a graph G is a proper total coloring of G using k colors such that any two adjacent vertices have different sets of colors. It was known that every planar graph G with $$\Delta \ge 10$$ is neighbor-distinguishing totally $$(\Delta +3)$$-colorable. In this paper, we extend this result to the case $$\Delta =9$$. Namely, we prove that every planar graph G with $$\Delta =9$$ is neighbor-distinguishing totally 12-colorable.