Hamilton cycles and paths in butterfly graphs

Abstract

AbstractA 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, The n 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.