Hamilton cycles and paths in butterfly graphs

A cycle C in a graph G is a Hamilton cycle if C contains every vertex of G. Similarly, a path P in G is a Hamilton path if P contains every vertex of G. We say that G is Hamilton-connected if for any pair of vertices, u and v of G, there exists a Hamilton path from u to v. If G is a bipartite graph with bipartition sets of equal size, and there is a Hamilton path from any vertex in one bipartition set to any vertex in the other, then G is said to be Hamilton-laceable. We present a proof showing that the n-dimensional k-ary butterfly graph, denoted BF (k, n), contains a Hamilton cycle. Then, we use this result in proving the stronger result that BF (k, n) is Hamilton-laceable when n is even and Hamilton-connected for odd values of n.