On almost 1-extendable graphs

A graph G is 1-extendable or almost 1-extendable if every edge is contained in a perfect or almost perfect matching of G, respectively. Let d ≥ 3 be an integer, and let G be a graph of order n with exactly one odd component such that the degree of each vertex is either d or d + 1. If G is not almost 1-extendable, then we prove that n ≥ 2d + 5. In the special case that d ≥ 4 is even and G is a d-regular graph, we obtain the better bound n ≥ 3d+ 5. Examples will show that the given bounds are best possible. We shall assume that the reader is familiar with standard terminology on graphs (see, e.g., Chartrand and Lesniak [2]). In this paper, all graphs are finite and simple. The vertex set of a graph G is denoted by V (G). The neighborhoodNG(x) = N(x) of a vertex x is the set of vertices adjacent with x, and the number dG(x) = d(x) = |N(x)| is the degree of x in the graph G. If d ≤ dG(x) ≤ d+ k for each vertex x in a graph G, then we speak of a close to regular graph or more precisely of a (d, d + k)-graph. If k = 0, then the graph is d-regular. If X is a subset of the vertex set of a graph G, then G[X] is the subgraph induced by X. A perfect matching (almost perfect matching) of a graph G, is a matching M in G with the property that every vertex (with exactly one exception) is incident with an edge of M . We denote by Kn the complete graph of order n and by Kr,s the complete bipartite graph with partite sets A and B, where |A| = r and |B| = s. If G is a graph and A ⊆ V (G), then we denote by q(G−A) the number of odd components in the subgraph G− A. A graph G is p-extendable if it contains a set of p independent edges and every set of p independent edges can be extended to a perfect matching. In 1980, Plummer [9] studied the properties of p-extendable graphs. We call a graph G almost p-extendable if it contains a set of p independent edges and every set of p independent edges can be extended to an almost perfect matching.