Overfull Conjecture for graphs with maximum degree 4

Abstract

Let $G$ be a simple graph with maximum degree $Δ(G)$. The graph $G$ is overfull if $\left|E(G)\right|> Δ(G)\lfloor |V(G)|/2\rfloor$. In 1986, Chetwynd and Hilton proposed the 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 $Δ(H)=Δ(G)$. In this paper, we give a proof of this conjecture for graphs with maximum degree $4$.