Shi–Shan conjecture for $(P_2\cup kP_1)$-free graphs

Let $P_b$ denote the path on $b$ vertices, and let $P_2\cup kP_1$ be the disjoint union of a two-vertex path and $k$ isolated vertices. A graph is $t$-tough if $|S|\geq t\,c(G-S)$ for every vertex set $S$ such that $G-S$ has at least two components; it is $r$-connected if deleting fewer than $r$ vertices leaves it connected. A graph is $(P_2\cup kP_1)$-free if it has no induced subgraph isomorphic to $P_2\cup kP_1$.

Shi–Shan conjecture. Let $k\geq 4$ be an integer and let $G$ be a $1$-tough and $2k$-connected $(P_2\cup kP_1)$-free graph. Then $G$ is hamiltonian.

The paper proves the analogous assertion with $4$-tough in place of $1$-tough, so the conjecture proposes that the toughness hypothesis can be weakened. The conjectured $1$-tough result remains open in the source.