AI 中文总结
本文针对最大度为4的图，对1986年切特温德和希尔顿提出的过满猜想进行证明，该猜想指出特定条件下简单图是二类图当且仅当包含特定过满子图，此为相关研究提供了重要结论。
AI 中文摘要
设\(G\)为最大度\(\Delta(G)\)的简单图。若\(\left|E(G)\right|> \Delta(G)\lfloor |V(G)|/2\rfloor\)，则\(G\)是过满的。1986年，切特温德和希尔顿提出过满猜想：若\(G\)是\(\Delta(G)>\frac{|V(G)|}{3}\)的简单图，那么\(G\)是二类图当且仅当\(G\)包含一个\(\Delta(H)=\Delta(G)\)的过满子图\(H\)。本文给出了最大度为4的图的该猜想的证明。
英文摘要
Let $G$ be a simple graph with maximum degree $Δ(G)$ and chromatic index $χ'(G)$. A graph $G$ is called edge-chromatic $Δ$-critical if $χ'(G)=Δ(G)+1$ and $χ'(H)< χ'(G)$ for every proper subgraph $H$ of $G$, and $G$ is overfull if $\left|E(G)\right|>Δ(G)\lfloor |V(G)|/2\rfloor$. In 1986, Chetwynd and Hilton proposed the influential Overfull Conjecture: If $G$ is a simple graph with $Δ(G)>\frac{|V(G)|}{3}$, then $G$ is a Class $2$ graph if and only if $G$ contains an overfull subgraph $H$ with $Δ(H)=Δ(G)$. Motivated by the structural analysis for $4$-critical graphs (SIAM J. Discrete Math. 2019), we show more properties in this paper, especially four new forbidden configurations in any $4$-critical graph, and provide a new structural proof of Overfull Conjecture for graphs with maximum degree $4$.
