Vertex-disjoint triangles in -free graphs with minimum degree at least

A graph is said to be K 1 , t -free if it does not contain an induced subgraph isomorphic to K 1 , t . Let h ( t , k ) be the smallest integer m such that every K 1 , t -free graph of order greater than m and with minimum degree at least t contains k vertex-disjoint triangles. In this paper, we obtain a lower bound of h ( t , k ) by a constructive method. According to the lower bound, we totally disprove the conjecture raised by Hong Wang [H. Wang, Vertex-disjoint triangles in claw-free graphs with minimum degree at least three, Combinatorica 18 (1998) 441–447]. We also obtain an upper bound of h ( t , k ) which is related to Ramsey numbers R ( 3 , t ) . In particular, we prove that h ( 4 , k ) = 9 ( k − 1 ) and h ( 5 , k ) = 14 ( k − 1 ) .