Longest Cycles in 2-Connected Quasi-Claw-Free Graphs

Abstract:

      A graph $G$ is called quasi-claw-free if it satisfies the property: $d(x,y)=2\Rightarrow$ there exists a vertex $u\in N(x)\cap N(y)$ such that $N[u]\subseteq N[x]\cup N[y]$. In this paper, we show that every 2-connected quasi-claw-free graph of order $n$ with $G\notin \mathcal{F}$ contains a cycle of length at least min$\{3\delta+2,n\}$, where $\mathcal{F}$ is a family of graphs.