The graph of the generating d-tuples of a finite soluble group and the swap conjecture

For a d-generated group G we consider the graph Lambda(d,G) in which the vertices are the ordered generating d-tuples and in which two vertices (x, ... , xd) and (y1, ..., yd) are adjacent if and only if there exists I subset of {1, ..., d} such that vertical bar I vertical bar >= [d/2] and xi= yi for each i is an element of I. We prove that if G is a finite soluble, then Lambda(d,G) is connected. We consider also the "swap graph" Delta(d,G) in which two generating d-tuples are adjacent if they differ only by one entry, proving the following: if G is an arbitrary finite group and d >= d(G) + 1, then Delta(d,G) is connected.