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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完全着色猜想的证明

图 G 的 $total\ coloring$ 是一个映射 $f:V(G) \cup E(G) \rightarrow \mathcal{K}$，其中 $\mathcal{K}$ 是一组颜色，满足以下三个条件： 1. $f(u) \neq f(v)$ 对于任意两个相邻顶点$u, v \in V(G)$；2. $f(e) \neq f(e')$ 任意两条相邻边$e, e' \in E(G)$; 和 3. $f(v) \neq f(e)$ 对于任何顶点 $v \in V(G)$ 和任何与同一顶点 $v$ 相关的边 $e \in E(G)$。$total\ chroming\ number$, $\chi''(G)$ 是$G$ 的$total\coloring$ 所需的最小颜色数。Behzad 和 Vizing 独立推测，对于任何图 $G$，$\chi''(G)\leq \Delta + 2$。这是经典的数学未解决问题之一。在本文中，我们通过证明 $total\chromo\number$$\chi'' 来解决这个经典猜想 图的 (G)$ 确实以 $\Delta+2$ 为界。我们的新方法涉及有限域 $\mathbb{Z}_p$ 上的代数设置和组合 nullstellensatz 的应用。