A Hamiltonian Result on $L_1$-Graphs

Abstract

A graph $G$ is called an $L_1$-graph if, for each triple of vertices $x,y,$ and $z$ with $d(x,y)=2$ and $z\in N(x)\cap N(y)$, $d(x)+d(y)\ge |N(x)\cup N(y)\cup N(z)|–1$. Let $G$ be a $3$-connected $L_1$-graph of order $n\ge 18$. If $\delta (G)\ge n/3$, then every pair of vertices $u$ and $v$ in $G$ with $d(u,v)\ge 3$ is connected by a Hamiltonian path of $G$.