The chvátal-erdo˝s condition for cycles in triangle-free graphs Original Research Article

It is proved that if G is a triangle-free graph with v vertices whose independence number does not exceed its connectivity then G has cycles of every length n for 4 ⩽ n ⩽ v(G) unless G = Kv/2,v/2 or G is a 5-cycle. This was conjectured by Amar, Fournier and Germa.