A Fan-type condition for cycles in $1$-tough and $k$-connected $(P_2\cup k{P}_1)$-free graphs

Zhiquan Hu, Jie Wang, Changlong Shen

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 $\{u_1,\dots ,u_k\}$ of $G$ suc…