Closure, path-factors and path coverings in claw-free graphs (Q2713641)

A graph \(G\) is said to be covered by subgraphs \(H_1,\ldots ,H_k\) if \(V(H_1)\cup \cdots \cup V(H_k)=V(G)\). A subgraph of \(G\) consisting of vertex disjoint paths covering \(G\) is called a path-factor of \(G\). Let \(G\) be a claw-free graph, let \(\text{cl}(G)\) be the closure of \(G\) (introduced by the reviewer in [J. Comb. Theory, Ser. B 70, No.~2, 217-224 (1997; Zbl 0872.05032)]) and let \(k\) be a positive integer. Then (i) \(G\) has a path-factor with \(k\) components if and only if \(\text{cl}(G)\) has a path-factor with \(k\) components, and (ii) \(G\) is covered by \(k\) paths if and only if \(\text{cl}(G)\) is covered by \(k\) paths.