arXiv - CS - Discrete Mathematics Pub Date : 2020-03-21 , DOI: arxiv-2003.09658
T Srinivasa Murthy

A $total\ coloring$ 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 same vertex $v$. The $total\ chromatic\ number$, $\chi''(G)$, is the minimum number of colors required for a $total\ coloring$ of $G$. Behzad and Vizing independently conjectured that, for any graph $G$, $\chi''(G)\leq \Delta + 2$. This is one of the classic mathematical unsolved problems. In this paper, we settle this classical conjecture by proving that the $total\ chromatic\ number$ $\chi''(G)$ of a graph is indeed bounded above by $\Delta+2$. Our novel approach involves algebraic settings over finite field $\mathbb{Z}_p$ and application of combinatorial nullstellensatz.

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