A Note on Cycle Lengths in Graphs

 We prove that for every c>0 there exists a constant K = K(c) such that every graph G with n vertices and minimum degree at least cn contains a cycle of length t for every even t in the interval [4,ec(G) − K] and every odd t in the interval [K,oc(G) − K], where ec(G) and oc(G) denote the length of the longest even cycle in G and the longest odd cycle in G respectively. We also give a rough estimate of the magnitude of K.