Number of disjoint 5-cycles in graphs

Let k >= 1, l >= 3 and s >= 5 be integers. In 1990, Eras and Faudree conjectured that if C is a graph of order 4k with delta(G) >= 2k, then G contains k vertex-disjoint 4-cycles. In this paper, we consider an analogous question for 5-cycles; that is to say if C is a graph of order 5k with delta(G) >= 3k, then G contains k vertex-disjoint 5-cycles? In support of this question, we prove that if C is a graph of order 5l with sigma(2) (G) > 6l - 2, then, unless (K(l - 2)$) over bar + K(2l+1),(2l+1) subset of G subset of K(l - 2) + K(2l+1),(2l+1), G contains l - 1 vertex-disjoint 5-cycles and a path of order 5, which is vertex-disjoint from the 1 1 5-cycles. In fact, we prove a more general result that if G is a graph of order 5k + 2s with sigma(2)(G) >= 6k + 2s, then, unless (K) over bar (k) + K(2k+s),(2k+s) subset of G subset of K(k) + K(2k+s), 2(k+s), G contains k +1 vertex-disjoint 5-cycles and a path of order 2s 5, which is vertex-disjoint from the k +1 5-cycles. As an application of this theorem, we give a short proof for determining the exact value of ex(n, (k + 1)C(5)), and characterize the extremal graph.