A Fan-type condition for cycles in $1$-tough and $k$-connected $(P_2\cup
kP_1)$-free graphs
A Fan-type condition for cycles in $1$-tough and $k$-connected $(P_2\cup
kP_1)$-free graphs
For a graph $G$, let $\mu_k(G):=\min~\{\max_{x\in S}d_G(x):~S\in \mathcal{S}_k\}$, where $\mathcal{S}_k$ is the set consisting of all independent sets $\{u_1,\ldots,u_k\}$ of $G$ such that some vertex, say $u_i$ ($1\leq i\leq k$), is at distance two from every other vertex in it. A graph $G$ is $1$-tough if for each cut set …