On a leverage problem in the hypercube

Abstract

The leverage of a set S of elements (vertices and edges) of a graph G, with respect to a graphical parameter P, is the change induced in P by the removal of S. We consider the case in which G is a hypercube and P is the sum of the distances between vertices. The determination of the minimum leverage of any set of k edges leads to the question of the existence of a perfect matching in which no two edges lie on a 4-cycle. We give a constructive proof of the existence of such a matching and note additional interesting problems.