ON MAXIMAL MATCHINGS OF CONNECTED GRAPHS

Abstract

Let I with |I| = k be a matching of a graph G (briefly, I is called a k-matching). If I is not a proper subset of any other matching of G, then I is a maximal k-matching and m(gk,G) is used to denote the number of maximal k-matchings of G. Let gk be a k-matching of G, if there exists a subset {e1, e2, …, ei} of E(G) \\ gk, i ≥ 1, such that (1) for any j ɛ {1, 2, …, i}, gk + {ej} is a (k + 1)-matching of G; (2) for any f ɛ E(G) \\ (gk ∪ {e1, e2, …, ei}), gk + {f} is not a matching of G; then gk is called an i wings k-matching of G and mi(gk, G) is used to denote the number of i wings k-matchings of G. In this paper, it is proved that both mi(gk,G) and m(gk, G) are edge reconstructible for every connected graph G, and as a corollary, it is shown that the matching polynomial is edge reconstructible.