Overfull Conjecture for graphs with maximum degree 4

Let $G$ be a simple graph with maximum degree $\mathrm{\Delta }(G)$. The graph $G$ is overfull if $\mid E(G)\mid >\mathrm{\Delta }(G)\lfloor \mathrm{\mid }V(G)\mathrm{\mid }\mathrm{/}2\rfloor$. In 1986, Chetwynd and Hilton proposed the Overfull Conjecture: If $G$ is a simple graph with $\mathrm{\Delta }(G)>\frac{\mathrm{\mid }V(G)\mathrm{\mid }}{3}$, then $G$ is a Class $2$ graph if and only if $G$ contains an overfull subgraph $H$ with $\mathrm{\Delta }(H)=\mathrm{\Delta }(G)$. In this paper, we give a proof of this conjecture for graphs with maximum degree $4$.