A related problem on $s$-Hamiltonian line graphs

A graph $G$ is said to be claw-free if $G$ does not contain $K_{1,3}$ as an induced subgraph. For an integer $s\ge 0$, $G$ is $s$-Hamiltonian if for any vertex subset $S\subset V(G)$ with $|S|\le s$, $G-S$ is Hamiltonian. Lai et al. in [On $s$-Hamiltonian line graphs of claw-free graphs, Discrete Math., 342 (2019)] proved that for a connected claw-free graph $G$ and any integer $s\ge 2$, its line graph $L(G)$ is $s$-Hamiltonian if and only if $L(G)$ is $(s+2)$-connected.

Motivated by above result, we in this paper propose the following conjecture. Let $G$ be a claw-free connected graph such that $L(G)$ is 3-connected and let $s\ge 1$ be an integer. If one of the following holds:

( $i$) $s\in \{1,2,3,4\}$ and $L(G)$ is essentially $(s+3)$-connected,

( $ii$) $s\ge 5$ and $L(G)$ is essentially $(s+2)$-connected,

then for any subset $S\subseteq V(L(G))$ with $|S|\le s$, $|D_{\le 1}(L(G)-S)|\le \lfloor \frac{s}{2}\rfloor$ and $L(G)-S-D_{\le 1}(L(G)-S)$ is Hamiltonian. Here, $D_{\le 1}(L(G)-S)$ denotes the set of vertices of degree at most 1 in $L(G)-S$. Furthermore, we in this paper deal with the cases $s\in \{1,2,3,4\}$ and $L(G)$ is essentially $(s+3)$-connected about this conjecture.