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

TL;DR

This paper proves that the Total Coloring Conjecture holds for 1-toroidal graphs with maximum degree at least 11 and no adjacent triangles, extending understanding of total colorings in complex graph classes.

Contribution

It establishes the TCC for 1-toroidal graphs with high maximum degree and no adjacent triangles, a new result in graph coloring theory.

Findings

TCC holds for 1-toroidal graphs with Δ ≥ 11 and no adjacent triangles

Total coloring can be achieved with at most Δ + 2 colors in these graphs

Extends total coloring results to a broader class of graphs

Abstract

A {\em 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 {\em total chromatic number} of a graph $G$, denoted by $\chiup''(G)$, is the minimum number of colors in a total coloring of $G$. The well-known Total Coloring Conjecture (TCC) says that every graph with maximum degree $\Delta$ admits a total coloring with at most $\Delta + 2$ colors. A graph is {\em $1$-toroidal} if it can be drawn in 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 $\Delta$ at least~$11$ and without adjacent triangles, then $G$ admits a total coloring with at most $\Delta + 2$ colors.