Closure and Hamiltonian-connectivity of claw-free graphs
In Ryjáček (1997), the closure cl(G) for a claw-free graph G is defined, and it is proved that G is hamiltonian if and only if cl(G) is hamiltonian. On the other hand, there exist infinitely many claw-free graphs G such that G is not hamiltonian-connected (resp. homogeneously traceable) while cl(G) is hamiltonian-connected (resp. homogeneously traceable). In this paper we define a new closure clk(G) (k≥1) as a generalization of cl(G) and prove the following theorems. (1)A claw-free graph G is hamiltonian-connected if and only if cl3(G) is hamiltonian-connected. (2) A claw-free graph G is homogeneously traceable if and only if cl2(G) is homogeneously traceable. We also discuss the uniqueness of the closure. © 1999 Elsevier Science B.V. All rights reserved.
Bollobás, B., Riordan, O., Ryjáček, Z., Saito, A., & Schelp, R. (1999). Closure and Hamiltonian-connectivity of claw-free graphs. Discrete Mathematics, 195 (1-3), 67-80. https://doi.org/10.1016/S0012-365X(98)00165-4