Two Results on the Hamiltonicity of $L_1$ – Graphs

Abstract

A graph $G$ is called an $L_1$-graph if, for each triple of vertices $u$, $v$, and $w$ with $d(u,v)=2$ and $w\in N(u)\cap N(v)$, the condition $d(u)+d(v)>|N(u)\cup N(v)\cup N(w)|–1$ holds. This paper presents two results on the hamiltonicity of $L_1$-graphs.