Disjoint Cliques in Claw-free Graphs

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≥3 s +4（k-s） and d（x）+ d（y）≥n-2 s +2 k +1 for any pair of non-adjacent vertices x, y of G, then G contains s disjoint K3 s and k-s disjoint K4 s such that all of them are disjoint. Moreover, the degree condition is sharp in some cases.