A Fan-type condition for cycles in 1-tough and k-connected (P2 ∪ kP1)-free graphs

For a graph G, define ${\mu }_k\left(G\right):=\min \{{\max }_{x\in S}{d}_G(x):S\in S_k\}$, where $S_k$ is the set consisting of all independent sets $\{u_1,\dots ,u_k\}$ of G such that some vertex, say $u_i$ ($1\le i\le k$), is at distance two from every other vertex in it. A graph G is called 1-tough if for each cut set $S\subseteq V\left(G\right)$, $G-S$ has no more than $\left|S\right|$ components. Recently, Shi and Shan [19] conjectured that for each integer $k\ge 4$, being 2k-connected is sufficient for 1-tough $(P_2\cup k{P}_1)$-free graphs to be hamiltonian, which was confirmed by Xu et al. [20] and Ota and Sanka [16], respectively. In this article, we generalize the above results through the following Fan-type theorem: If G is a 1-tough and k-connected $(P_2\cup k{P}_1)$-free graph and satisfies ${\mu }_{k+1}\left(G\right)\ge \frac{7k-6}{5}$, where $k\ge 2$ is an integer, then G is hamiltonian or the Petersen graph.