On the Perfect Matchings of Near Regular Graphs

Abstract

Let k, h be positive integers with k ≤ h. A graph G is called a [k, h]-graph if k ≤ d(v) ≤ h for any ${v \in V(G)}$. Let G be a [k, h]-graph of order 2n such that k ≥ n. Hilton (J. Graph Theory 9:193–196, 1985) proved that G contains at least ${\lfloor k/3\rfloor}$ disjoint perfect matchings if h = k. Hilton’s result had been improved by Zhang and Zhu (J. Combin. Theory, Series B, 56:74–89, 1992), they proved that G contains at least ${\lfloor k/2\rfloor}$ disjoint perfect matchings if k = h. In this paper, we improve Hilton’s result from another direction, we prove that Hilton’s result is true for [k, k + 1]-graphs. Specifically, we prove that G contains at least ${\lfloor\frac{n}3\rfloor+1+(k-n)}$ disjoint perfect matchings if h = k + 1.