TY - JOUR

T1 - On s-hamiltonian line graphs of claw-free graphs

AU - Lai, Hong Jian

AU - Zhan, Mingquan

AU - Zhang, Taoye

AU - Zhou, Ju

N1 - Funding Information:
The research is partially supported by National Natural Science Foundation of China grants (Nos. 11771039 and 11771443). The authors are indebted to an anonymous reviewer for providing insightful comments which helped to improve the manuscript. The authors declared that they had no conflicts of interest with respect to their authorship or the publication of this article.

PY - 2019/11

Y1 - 2019/11

N2 - For an integer s≥0, a graph G is s-hamiltonian if for any vertex subset S⊆V(G) with |S|≤s, G−S is hamiltonian, and G is s-hamiltonian connected if for any vertex subset S⊆V(G) with |S|≤s, G−S is hamiltonian connected. Thomassen in 1984 conjectured that every 4-connected line graph is hamiltonian (see Thomassen, 1986), and Kučzel and Xiong in 2004 conjectured that every 4-connected line graph is hamiltonian connected (see Ryjáček and Vrána, 2011). In Broersma and Veldman (1987), Broersma and Veldman raised the characterization problem of s-hamiltonian line graphs. In Lai and Shao (2013), it is conjectured that for s≥2, a line graph L(G) is s-hamiltonian if and only if L(G) is (s+2)-connected. In this paper we prove the following. (i) For an integer s≥2, the line graph L(G) of a claw-free graph G is s-hamiltonian if and only if L(G) is (s+2)-connected. (ii) The line graph L(G) of a claw-free graph G is 1-hamiltonian connected if and only if L(G) is 4-connected.

AB - For an integer s≥0, a graph G is s-hamiltonian if for any vertex subset S⊆V(G) with |S|≤s, G−S is hamiltonian, and G is s-hamiltonian connected if for any vertex subset S⊆V(G) with |S|≤s, G−S is hamiltonian connected. Thomassen in 1984 conjectured that every 4-connected line graph is hamiltonian (see Thomassen, 1986), and Kučzel and Xiong in 2004 conjectured that every 4-connected line graph is hamiltonian connected (see Ryjáček and Vrána, 2011). In Broersma and Veldman (1987), Broersma and Veldman raised the characterization problem of s-hamiltonian line graphs. In Lai and Shao (2013), it is conjectured that for s≥2, a line graph L(G) is s-hamiltonian if and only if L(G) is (s+2)-connected. In this paper we prove the following. (i) For an integer s≥2, the line graph L(G) of a claw-free graph G is s-hamiltonian if and only if L(G) is (s+2)-connected. (ii) The line graph L(G) of a claw-free graph G is 1-hamiltonian connected if and only if L(G) is 4-connected.

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U2 - 10.1016/j.disc.2019.06.006

DO - 10.1016/j.disc.2019.06.006

M3 - Article

AN - SCOPUS:85067928771

VL - 342

SP - 3006

EP - 3016

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 11

ER -