Adjacent vertex distinguishing total colorings of outer 1-planar graphs *

An adjacent vertex distinguishing total coloring of a graph G is a proper total coloring of G such that no pair of adjacent vertices are incident to the same set of colors. The minimum number of colors required for an adjacent vertex distinguishing total coloring of G is denoted by Χ'a(ß). In this paper, we prove that if G is an outer 1-planar graph with at least two vertices, then Χ'a(G) ≤ max { Δ + 2, 8 } . Moreover, we also prove that when Δ ≤ 7, Χ'a(G) = Δ + 2 if and only if G contains two adjacent vertices of maximum degree. © 2019 Charles Babbage Research Centre. All rights reserved.