Large Neighbourhood Unions and Edge-disjoint Perfect Matchings

We show that if, for any fixed \(r\), the neighbourhood unions of all \(r\)-sets of vertices are large enough, then \(G\) will have many edge-disjoint perfect matchings. In particular, we show that given fixed positive integers \(r\) and \(c\) and a graph \(G\) of even order \(n\), if the minimum degree is at least \(r + c – 1\) and if the neighbourhood union of each \(r\)-set of vertices is at least \(n/2 + \left(2\lfloor\frac{(c + 1)}{2}\rfloor – 1\right)r\), then \(G\) has \(c\) edge-disjoint perfect matchings, for \(n\) large enough. This extends earlier work by Faudree, Gould and Lesniak on neighbourhood unions of pairs of vertices.