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For an integer s ≥ 0, a graph G is s‐hamiltonian if for any vertex subset with |S′| ≤ s, G ‐ S′ is hamiltonian. It is well known that if a graph G is s‐hamiltonian, then G must be (s+2)‐connected. The converse is not true, as there exist arbitrarily highly connected nonhamiltonian graphs. But for line graphs, we prove that when s ≥ 5, a line graph is s‐hamiltonian if and only if it is (s+2)‐connected.
For integers k,s with 0⩽k⩽s⩽|V(G)|-3, a graph G is called s-Hamiltonian if the removal of any k vertices results in a Hamiltonian graph. For a simple connected graph that is not a path, a cycle or a K1,3 and an integer s⩾0, we define hs(G)=min{m:Lm(G)iss-Hamiltonian} and l(G)=max{m:G has a divalent path of length m that is not both of length 2 and in a K3}, where a divalent path in G is a non-closed...
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