Edge-chromatic $4$-critical graphs and Overfull Conjecture for graphs with maximum degree $4$

Abstract

Let $G$ be a simple graph with maximum degree $\hat{I}”(G)$ and chromatic index $\ddot{I}{‡}^{\prime }(G)$. A graph $G$ is called edge-chromatic $\hat{I}”$-critical if $\ddot{I}{‡}^{\prime }(G)=\hat{I}”(G)+1$ and $\ddot{I}{‡}^{\prime }(H)<\ddot{I}{‡}^{\prime }(G)$ for every proper subgraph $H$ of $G$, and $G$ is overfull if $\left|E(G)\right|>\hat{I}”(G)\lfloor |V(G)|/2\rfloor$. In 1986, Chetwynd and Hilton proposed the influential Overfull Conjecture: If $G$ is a simple graph with $\hat{I}”(G)>\frac{|V(G)|}{3}$, then $G$ is a Class $2$ graph if and only if $G$ contains an overfull subgraph $H$ with $\hat{I}”(H)=\hat{I}”(G)$. 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 $4$.