A proof of the Total Coloring Conjecture

$\mathit{\text{Total}}\mathit{\text{Coloring}}$ 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 $f:V(G)\cup E(G)\to \mathcal{K}$, where $\mathcal{K}$ is a set of colors, satisfying the following three conditions: 1. $f(u)\mathrm{\ne }f(v)$ for any two adjacent vertices $u,v\in V(G)$; 2. $f(e)\mathrm{\ne }f(e^{\mathrm{\prime }})$ for any two adjacent edges $e,e^{\mathrm{\prime }}\in E(G)$; and 3. $f(v) \neq f(e)$foranyvertex$v \in V(G)$andanyedge$e \in E(G)$ that is incident to the same vertex $v$. The $\mathit{\text{total}}\mathit{\text{chromatic}}\mathit{\text{number}}$, ${\chi }^{\mathrm{\prime }\mathrm{\prime }}(G)$, is the minimum number of colors required for a \textit{total coloring} of $G$. Behzad (1965), and Vizing (1968), conjectured that for any graph $G$ ${\chi }^{\mathrm{\prime }\mathrm{\prime }}(G)\le \mathrm{\Delta }+2$. This conjecture is one of the classic unsolved mathematical problems. In this paper, we settle this classical conjecture by proving that the $\mathit{\text{total}}\mathit{\text{chromatic}}\mathit{\text{number}}$ ${\chi }^{\mathrm{\prime }\mathrm{\prime }}(G)$ of a graph is indeed bounded above by $\mathrm{\Delta }+2$. 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.