On graph closures

Ryjacek (J. Combin. Theory B 70 (1997) 217) introduced a very useful notion of a closure cl(G) for a claw-free graph G and proved, in particular, that c(G)=c(cl(G)) where c(H) is the length of a longest cycle in H. In this paper, we describe some strengthenings of the main results in Ryjacek (J. Combin. Theory B 70 (1997) 217). As a result, we introduce some new closures a(G) that can be used in a wider class of graphs, and show, in particular, that c(G) = c(sigma(G)) and P(G) = p(sigma(G)) where p(H) is the length of a longest path in H. As a byproduct, we give some new sufficient conditions for graphs to have a Hamiltonian cycle, path, v-path, and uv-path, and show, in particular, that every claw-free 9-connected graph is Hamiltonian connected. We also give a construction that provides infinitely many counterexamples to the conjecture on the so-called cl(2)-closure in Bollobas et al. (C) 2002 Elsevier B.V. All rights reserved.