A related problem on $ s $-Hamiltonian line graphs

Abstract

<abstract><p>A graph $ G $ is said to be <italic>claw-free</italic> if $ G $ does not contain $ K_{1, 3} $ as an induced subgraph. For an integer $ s\geq0 $, $ G $ is <italic>$ s $-Hamiltonian</italic> if for any vertex subset $ S\subset V(G) $ with $ |S|\leq 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\geq 2 $, its line graph $ L(G) $ is $ s $-Hamiltonian if and only if $ L(G) $ is $ (s+2) $-connected.</p> <p>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\geq1 $ be an integer. If one of the following holds:</p> <p>($ i $) $ s\in\{1, 2, 3, 4\} $ and $ L(G) $ is essentially $ (s+3) $-connected,</p> <p>($ ii $) $ s\geq5 $ and $ L(G) $ is essentially $ (s+2) $-connected,</p> <p>then for any subset $ S\subseteq V(L(G)) $ with $ |S|\leq s $, $ |D_{\leq1}(L(G)-S)|\leq\left \lfloor \frac{s}{2} \right \rfloor $ and $ L(G)-S-D_{\leq1}(L(G)-S) $ is Hamiltonian. Here, $ D_{\leq1}(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.</p></abstract>