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 Λd(G) in which the vertices are the ordered generating d-tuples and in which two vertices (x1,…,xd) and (y1,…,yd) are adjacent if and only if there exists I⊆{1,…,d} such that |I|⩾[d/2] and xi=yi for each i∈I. We prove that if G is a finite soluble, then Λd(G) is connected. We consider also the “swap graph” Δd(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 Δd(G) is connected.