Adjacent vertex distinguishing total coloring of planar graphs with maximum degree 9

Let k be a positive integer. An adjacent vertex distinguishing (for short, AVD) total-k-coloring of a graph G is a proper total-k-coloring of G such that any two adjacent vertices have different color sets, where the color set of a vertex nu contains the color of nu and the colors of its incident edges. It was conjectured that any graph with maximum degree Delta has an AVD total-(Delta+3)-coloring. The conjecture was confirmed for any graph with maximum degree at most 4 and any planar graph with maximum degree at least 10. In this paper, we verify the conjecture for all planar graphs with maximum degree at least 9. Moreover, we prove that any planar graph with maximum degree at least 10 has an AVD total-(Delta + 2)coloring and the bound Delta + 2 is sharp. (C) 2019 Elsevier B.V. All rights reserved.