Closure and Hamiltonian-connectivity of claw-free graphs (Q1296972)

The authors define for a graph \(G\), a new closure \(\text{cl}_k(G)\) \((k\geq 1)\) as a generalization of \(\text{cl}(G)\) and prove the following results: (1) A claw-free graph \(G\) is Hamiltonian-connected iff \(\text{cl}_3(G)\) is Hamiltonian-connected. (2) A claw-free graph \(G\) is homogeneously traceable iff \(\text{cl}_2(G)\) is homogeneously traceable. The uniqueness of the closure has also been discussed. Finally, they make the following conjecture: A claw-free graph \(G\) is Hamiltonian-connected iff \(\text{cl}_2(G)\) is Hamiltonian-connected.