Vertex-disjoint copies of K1,3K1,3 in K1,rK1,r-free graphs

A graph GG is said to be K1,rK1,r-free if GG does not contain an induced subgraph isomorphic to K1,rK1,r. Let kk, rr be integers with k≥2k≥2, r≥4r≥4. In this paper, we prove that if GG is a K1,rK1,r-free graph of order at least (k−1)(3r−2)+1(k−1)(3r−2)+1 with δ(G)≥3δ(G)≥3, then GG contains kk vertex-disjoint copies of K1,3K1,3. This result shows that Fujita’s conjecture (2008) is true for t=3t=3 and r≥4r≥4.