A proof of the Total Coloring Conjecture

Abstract: \textit{Total Coloring} of a graph is a major coloring problem in combinatorial mathematics, introduced in the early $1960$s. A \textit{total coloring} of a graph $G$ is a map $f:V(G) \cup E(G) \rightarrow \mathcal{K}$, where $\mathcal{K}$ is a set of colors, satisfying the following three conditions: 1. $f(u) \neq f(v)$ for any two adjacent vertices $u, v \in V(G)$; 2. $f(e) \neq f(e')$ for any two adjacent edges $e, e' \in E(G)$; and 3. $f(v) \neq f(e)$ for any vertex $v \in V(G)$ and any edge $e \in E(G)$ that is incident to the same vertex $v$. The \textit{total chromatic number}, $\chi''(G)$, is the minimum number of colors required for a \textit{total coloring} of $G$. Behzad (1965), and Vizing (1968), conjectured that for any graph $G$ $\chi''(G)\leq \Delta + 2$. This conjecture is one of the classic unsolved mathematical problems. In this paper, we settle this classical conjecture by proving that the \textit{total chromatic number} $\chi''(G)$ of a graph is indeed bounded above by $\Delta+2$. Our novel approach involves algebraic settings over a finite field $\mathbb{Z}_p$ and Vizing's theorem is an essential part of the algebraic settings.