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On $s$-hamiltonian line graphs of claw-free graphs.
Faculty Author(s): Zhan, Mingquan
Student Author(s): -
Department: MATH
Publication: Discrete Mathematics
Year: 2019
Abstract: A graph is $s$-hamiltonian ($s$-hamiltonian connected) if the deletion of any set of up to $s$ vertices leaves a hamiltonian (hamiltonian connected) graph. H.-J. Lai and Y. Shao [J. Graph Theory {\bf 74} (2013), no.~3, 344--358; [msn] MR3105554 [/msn]] showed that, for $s\ge 5$, a line graph is $s$-hamiltonian if and only if it is ($s+2$)-connected. Here the authors show the line graph of a claw-free graph is 1-hamiltonian connected if and only if it is 4-connected and, for $s\ge 2$, it is $s$-hamiltonian if and only if it is ($s+2$)-connected. They conjecture, for $s\ge 2$, that a line graph is $s$-hamiltonian if and only if it is ($s+2$)-connected and a claw-free graph is $s$-hamiltonian if and only if it is ($s+2$)-connected, and, for $s\ge 1$, a line graph is $s$-hamiltonian connected if and only if it is ($s+3$)-connected and a claw-free graph is $s$-hamiltonian connected if and only if it is ($s+3$)-connected.
Link: On $s$-hamiltonian line graphs of claw-free graphs.