On the size of odd order graphs with no almost perfect matching

A graph G is a (d, d + 1)-graph if the degree of each vertex of G is either d or d + 1. If d ≥ 2 is an integer and G a (d, d + 1)-graph with exactly one odd component and with no almost perfect matching, then we show in this paper that |V (G)| ≥ 4(d + 1) + 1 and |V (G)| ≥ 4(d + 1) + 3 when d is odd. This result generalizes corresponding statements by C. Zhao (J. Combin. Math. Combin. Comput. 9 (1991), 195–198) and W.D. Wallis (Ars Combin. 11 (1981), 295–300) on the size of even order graphs without a perfect matching. Examples will show that the given bounds are best possible, and some related results are also presented. 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 and edge set of a graph are denoted by V (G) and E(G), respectively. The neighborhood NG(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. If d ≤ dG(x) ≤ d + k for each vertex x in a graphG, then we speak of a (d, d+k)-graph. IfM is a matching in a graph G with the property that every vertex (with exactly one exception) is incident with an edge of M , then M is a perfect matching (an almost perfect matching). We denote byKn the complete graph of order n and byKr,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. The proof of our main theorem is based on the following generalization of Tutte’s famous 1-factor theorem [4] by Berge [1] in 1958, and we call it the theorem of TutteBerge (for a proof see e.g., [5]). Theorem of Tutte-Berge (Berge [1] 1958) Let G be a graph of order n. If M is a maximum matching of G, then n− 2|M | = max A⊆V (G) {q(G−A)− |A|}.