Vertex-disjoint copies of K1+(K1∪K2)K1+(K1∪K2) in claw-free graphs

A graph G is said to be claw-free if G   does not contain an induced subgraph isomorphic to K1,3K1,3. Let k   be an integer with k⩾2k⩾2. We prove that if G   is a claw-free graph of order at least 7k-67k-6 and with minimum degree at least 3, then G contains k   vertex-disjoint copies of K1+(K1∪K2)K1+(K1∪K2).