Circumferences of $2$-Connected Quasi-Claw-Free Graphs

Abstract

A graph $G$ is quasi-claw-free if it satisfies the property: $d(x,y)=2\Rightarrow$ there exists $u\in N(x)\cap N(y)$ such that $N[u]\subseteq N[x]\cup N[y]$. In this paper, we prove that the circumference of a $2$-connected quasi-claw-free graph $G$ on $n$ vertices is at least $min\{3\delta +2,n\}$ or $G\in \mathcal{F}$, where $\mathcal{F}$ is a class of nonhamiltonian graphs of connectivity $2$. Moreover, we prove that if $n\le 40$, then $G$ is hamiltonian or $G\in \mathcal{F}$.