On a closure concept in claw-free graphs

Abstract

If G is a claw-free graph, then there is a graph cl(G) such that (i) G is a spanning subgraph of cl(G), (ii) cl(G) is a line graph of a triangle-free graph, and (iii) the length of a longest cycle in G and in cl(G) is the same. A sufficient condition for hamiltonicity in claw-free graphs, the equivalence of some conjectures on hamiltonicity in 2-tough graphs and the hamiltonicity of 7-connected claw-free graphs are obtained as corollaries. © 1997 Academic Press.