@article{Liu2022, 
author = {Xia Liu},
title = {A related problem on    s-Hamiltonian line graphs},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {10},
pages = {19553-19561},
keywords = {essentially, s-Hamiltonian, supereulerian, collapsible, dominating},
url = {https://www.sciopen.com/article/10.3934/math.20221073},
doi = {10.3934/math.20221073},
abstract = {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  ≥  0,    G is    s-Hamiltonian if for any vertex subset    S  ⊂  V  (  G  ) with        |    S      |    ≤  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  ≥  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  ≥  1 be an integer. If one of the following holds:(   i)    s  ∈  {  1  ,  2  ,  3  ,  4  } and    L  (  G  ) is essentially    (  s  +  3  )-connected,(   i  i)    s  ≥  5 and    L  (  G  ) is essentially    (  s  +  2  )-connected,then for any subset    S  ⊆  V  (  L  (  G  )  ) with        |    S      |    ≤  s,        |        D          ≤      1        (  L  (  G  )  −  S  )      |    ≤      ⌊          s      2        ⌋   and    L  (  G  )  −  S  −      D          ≤      1        (  L  (  G  )  −  S  ) is Hamiltonian. Here,        D          ≤      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  ∈  {  1  ,  2  ,  3  ,  4  } and    L  (  G  ) is essentially    (  s  +  3  )-connected about this conjecture.}
}