A (k; g)-graph is a k-regular graph with girth g. A (k; g)-cage is a(k; g)-graph with the least number of vertices. The order of a (k; g)-cageis denoted by f(k; g). In this paper we show that f(k +2; g)≥f(k; g)for k≥2 and present some partial results to support the conjecturethat f(k1; g) <f(k2; g) if k1<k2.

Among other results, we show that if for any given edge e of an r-regular graphG of even order, G has a 1-factor containing e, then G has a k-factor containing eand another one avoiding e for all k, 1≤k≤r−1.

Let G be a graph of order n≥4k+1, where k is a positive integer with kn even and δ(G)≥k. We prove that if the degree sum of each pair of nonadjacent ertices is at least n+1, then G has a k-factor including any given edge. Similarly, a sufficient condition for graphs to have a k-factor excluding any given edge is also given.

A graph G is k-extendable if it contains a set of k independent edges and each set of k independent edges can be extended to a perfect matching of G. In this note, we present degree-sum conditions for graphs to be k-extendable.

Let G be a graph with vertex set V(G). Let n; k and d be non-negative integers such that n+2k+d6|V(G)|−2 and |V(G)|−n−d is even. A matching which covers exactly |V(G)|−d vertices of G is called a defect-d matching of G. If when deleting any n vertices of G the remaining subgraph contains a matching of k edges and every k-matching can be extended o a defect-d matching, then G is called a (n; k; d)-graph. In this paper a characterization of (n; k; d)-graphs is given and several properties (such as connectivity, minimum degree, hierarchy, etc.) of (n; k; d)-graphs are investigated.