10.
For an integer
, a graph
is
-hamiltonian if for any vertex subset
with
,
is hamiltonian, and
is
-hamiltonian connected if for any vertex subset
with
,
is hamiltonian connected. Thomassen in 1984 conjectured that every 4-connected line graph is hamiltonian (see Thomassen, 1986), and Ku?zel and Xiong in 2004 conjectured that every 4-connected line graph is hamiltonian connected (see Ryjá?ek and Vrána, 2011). In Broersma and Veldman (1987), Broersma and Veldman raised the characterization problem of
-hamiltonian line graphs. In Lai and Shao (2013), it is conjectured that for
, a line graph
is
-hamiltonian if and only if
is
-connected. In this paper we prove the following.(i) For an integer
, the line graph
of a claw-free graph
is
-hamiltonian if and only if
is
-connected.(ii) The line graph
of a claw-free graph
is 1-hamiltonian connected if and only if
is 4-connected.
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