On the cycle space of a random graph.

Write C(G) for the cycle space of a graph G, C-kappa(G) for the subspace of C(G) spanned by the copies of the kappa-cycle C-kappa in G, T-kappa for the class of graphs satisfying C-kappa(G)=C(G), and Q(kappa) for the class of graphs each of whose edges lies in a C-kappa. We prove that for every odd kappa >= 3 and G=G(n,p), max(p) Pr(G is an element of Q kappa backslash T kappa) -> 0; so the C-kappa's of a random graph span its cycle space as soon as they cover its edges. For kappa = 3 this was shown in [6].