Triangles in claw-free graphs

Abstract

A graph is said to be claw-free if it does not contain an induced subgraph isomorphic to K 1,3. Let k be a positive integer. Our main result is as follows: If G is a claw-free graph of order at least 3 k and d(x) + d( y)⩾3 k + 1 for every pair of non-adjacent vertices x and y of G, then G contains k vertex-disjoint triangles unless either k is odd and G is one exceptional graph of order 3 k + 1, or k is 1 and G is the union of vertex-disjoint cycles of length at least 4.