论文标题
总体着色猜想的证明
A proof of the Total Coloring Conjecture
论文作者
论文摘要
图形的\ textit {总彩色}是组合数学中的主要着色问题，在$ 1960 $ s的早期中引入。图$ g $的a \ textit {总彩色}是地图$ f：v（g）\ cup e（g）\ rightarrow \ mathcal {k} $，其中$ \ mathcal {k} $是一组颜色，满足以下三个条件：1。 2。$ f（e）\ neq f（e'）$ for任何两个相邻边缘$ e，e'\ in E（g）$;和3。$ f（v）\ neq f（e）$对于v（g）$中的任何顶点$ v \ un E（g）$中的任何边缘$ e \ in（g）$中的任何边缘$ e \。 \ textIt {总色度}，$χ''（g）$，是$ g $的\ textit {Total Coloring}所需的最小颜色数。 Behzad（1965）和Viping（1968）猜想，对于任何图$ g $ $ $χ''（g）\leqΔ+ 2 $。这个猜想是经典的未解决的数学问题之一。在本文中，我们通过证明\ textIt {总的色度} $χ''（g）$确实以$δ+2 $限制了\ textit {总体色数} $χ''（g）$来解决这个经典的猜想。我们的新方法涉及在有限字段$ \ mathbb {z} _p $和Viping的定理上的代数设置，这是代数设置的重要组成部分。
\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}, $χ''(G)$, is the minimum number of colors required for a \textit{total coloring} of $G$. Behzad (1965), and Vizing (1968), conjectured that for any graph $G$ $χ''(G)\leq Δ+ 2$. 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} $χ''(G)$ of a graph is indeed bounded above by $Δ+2$. 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.
