Covering cycles in sparse graphs

Let k >= 2 be an integer. Kouider and Lonc proved that the vertex set of every graph G with n >= n0(k) vertices and minimum degree at least n/k can be covered by k-1 cycles. Our main result states that for every alpha>0 and p=p(n)is an element of(0,1], the same conclusion holds for graphs G with minimum degree (1/k+alpha)np that are sparse in the sense that eG(X,Y)<= p|X||Y|+o(np|X||Y|/log3n)for all X,Y subset of V(G). In particular, this allows us to determine the local resilience of random and pseudorandom graphs with respect to having a vertex cover by a fixed number of cycles. The proof uses a version of the absorbing method in sparse expander graphs.