A Fan-type condition for cycles in $1$-tough and k-connected $(P_2∪ kP_1)$-free graphs

For a graph G, let μₖ(G):=min~∈ SdG(x):~S∈ Sₖ\, where Sₖ is the set consisting of all independent sets ₁,…,uₖ\ of G such that some vertex, say uᵢ (1≤ i≤ k), is at distance two from every other vertex in it. A graph G is $1$-tough if for each cut set S⊆ V(G), $G-S$ has at most $|S|$ components. Recently, Shi and Shan {Shi} conjectured that for each integer k≥ 4, being $2k$-connected is sufficient for $1$-tough (P₂∪ kP₁)-free graphs to be hamiltonian, which was confirmed by Xu et al. {Xu} and Ota and Sanka {Ota2}, respectively. In this article, we generalize the above results through the following Fan-type theorem: Let k be an integer with k≥ 2 and let G be a $1$-tough and k-connected (P₂∪ kP₁)-free graph with μₖ₊₁(G)≥7k-6/5, then G is hamiltonian or the Petersen graph.