Total coloring of 1-toroidal graphs with maximum degree at least 11 and no adjacent triangles

A total coloring of a graph G is an assignment of colors to the vertices and the edges of G such that every pair of adjacent/incident elements receive distinct colors. The total chromatic number of a graph G, denoted by χ′′(G), is the minimum number of colors needed in a total coloring of G. The most well-known Total Coloring Conjecture (TCC) says that every graph with maximum degree ∆ admits a total coloring with at most ∆ + 2 colors. A graph is 1-toroidal if it can be drawn on torus such that every edge crosses at most one other edge. In this paper, we investigate the total coloring of 1-toroidal graphs, and prove that the TCC holds for the 1-toroidal graphs with maximum degree at least 11 and some restrictions on the triangles. Consequently, if G is a 1-toroidal graph with maximum degree ∆ at least 11 and without adjacent triangles, then G admits a total coloring with at most ∆ + 2 colors.