Hamiltonian Cycles in $N^2$-Locally Connected Claw-Free Graphs

Abstract

For a vertex $v$ in a graph $G$, we denote by $N^2(v)$ the set $(N_1(N_1(v))\setminus \{v\})\cup N_1(v)=\{x\in V(G):1\le d(x,v)\le 2\}$, where $d(x,v)$ denotes the distance between $x$ and $v$. A vertex $v$ is $N^2$-locally connected if the subgraph induced by $N^2(v)$ is connected. A graph $G$ is called $N^2$-locally connected if every vertex of $G$ is $N^2$-connected. A well-known result by Oberly and Sumner is that every connected locally connected claw-free graph on at least three vertices is Hamiltonian. This result was improved by Ryjacek using the concept of second-type neighborhood. In this paper, using the concept of $N^2$-locally connectedness, we show that every connected $N^2$-locally connected claw-free graph $G$ without vertices of degree $1$, which does not contain an induced subgraph $H$ isomorphic to one of $G_1,G_2,G_3$, or $G_4$, is Hamiltonian, hereby generalizing the result of Oberly and Sumner (J. Graph Theory, $3(1979)351-356$)and the result of $Ryjacek$( J. Graph Theory, $14(1990)$ 321-381)