Total coloring graphs with large minimum degree

TL;DR

This paper proves that large graphs with high minimum degree satisfy the Total Coloring Conjecture, establishing an upper bound on total chromatic number relative to maximum degree.

Contribution

It demonstrates that graphs with minimum degree exceeding half of the vertices meet the Total Coloring Conjecture, extending understanding of total coloring in dense graphs.

Findings

Graphs with minimum degree > (1/2 + ε)n satisfy the Total Coloring Conjecture

Total chromatic number is at most maximum degree + 2 for large dense graphs

Results hold for sufficiently large graphs with high minimum degree

Abstract

We prove that for all $\varepsilon>0$, there exists a positive integer $n_0$ such that if $G$ is a graph on $n\geq n_0$ vertices with $\delta(G)\geq\tfrac{1}{2}(1 + \varepsilon)n$, then $G$ satisfies the Total Coloring Conjecture, that is, $\chi_T(G)\leq \Delta(G)+2$.