Long Cycles Generate the Cycle Space of a Graph

Let G be a 2-connected graph in which the degree of every vertex is at least d. We prove that the cycles of length at least d + 1 generate the cycle space of G, unless G ≌ K_d+1 and d is odd. As a corollary, we deduce that the cycles of length at least d + 1 generate the subspace of even cycles in G. We also establish the existence of odd cycles of length at least d + 1 in the case when G is not bipartite.A second result states: if G is 2-connected with chromatic number at least k, then the cycles of length at least k generate the cycle space of G, unless G ≌ K_k and k is even. Similar corollaries follow, among them a stronger version of a theorem of Erdös and Hajnal.