It is shown that for any 4-regular graph G there is a collection ƒ of paths of length 4-such that each edge of G belongs to exactly two of the paths and each vertex of G occurs exactly twice as an endvertex of a path of ƒ This proves a special case of a conlecture of Bondy

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.

Let n be a non-negative integer. A graph G is said to be n-factor-critical if the subgraph G-S has a perfect matching for any subset S of V(G) with jSj=n. In this paper, we provide much shorter proofs of two main theorems about extendability and factor-criticality obtained by Favaron [2]. Furthermore, we present a solution to an open problem posted in the same paper.

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 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.

Let n be a non-negative integer. A graph G is said to be n-matchable if the subgraph G−S has a perfect matching for any subset S of V (G) with |S|=n. In this paper, we obtain sufficient conditions for different classes of graphs to be nmatchable. Since 2k-matchable graphs must be k-extendable, we have generalized the results about k-extendable graphs. All results in this paper are sharp.

Let G be a simple connected graph on 2n vertices with perfect matching. For a given positive integer k (0≤k≤n−1), G is k-extendable if any matching of size k in G is contained in a perfect matching of G. It is proved that if G is a k-extendable graph on 2n vertices with k≤n/2, then either G is bipartite or the connectivity of G is at least 2k. As a corollary, we show that if G is a maximal k-extendable graph on 2n vertices with n+2≤2k+1, then G is Kn; n if k+16≤n and G is K2n if 2k+16≤2n−1. Moreover, if G is a minimal k-extendable graph on 2n vertices with n+1≤2k+1 and k+16≤n thenthe minimum degree of G is k+1. We also discuss the relationship between the k-extendable graphs and the Hamiltonian graphs.

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.

A graph G is 2-extendable if any two independent edges of G are contained in a perfect matching of G, A Cayley graph of even order over an abelian group is 2-extendable if and only if it is not isomorphic to any of the following circulant graphs: (Ⅰ) Z2n (1, 2n-1), n≥3; (Ⅱ) Z2n (1, 2, 2n-1, 2n-2), n≥3; (Ⅲ) Z4n (1, 4n-1, 2n), n≥2; (Ⅳ) Z4n+2 (2, 4n, 2n+1), n≥1; and (Ⅴ) Z4n+2 (1, 4n+1, 2n, 2n+2), n≥1.