Menu
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 \(\chi”_a(G)\). In this paper, we prove that if \(G\) is an outer 1-planar graph with at least two vertices, then \(\chi”_a(G) \leq \max\{\Delta + 2, 8\}\). Moreover, we also prove that when \(\Delta \geq 7\), \(\chi”_a(G) = \Delta + 2\) if and only if \(G\) contains two adjacent vertices of maximum degree.