Disjoint cliques in claw-free graphs

Abstract

A graph is said to be claw-free if it does not contain an induced subgraph isomorphic to K1,3. Let s and k be two integers with 0 ≤ s ≤ k and let G be a claw-free graph of order n. In this paper, we investigate clique partition problems in claw-free graphs. It is proved that if n ≥ 3s+4(k−s) and d(x)+d(y) ≥ n−2s+2k+1 for any pair of non-adjacent vertices x, y of G, then G contains s disjoint K3s and k − s disjoint K4s such that all of them are disjoint. Moreover, the degree condition is sharp in some cases.