Hamilton decompositions of directed cubes and products

Abstract

Call a directed graph G ↔ symmetric if it is obtained from an undirected graph G by replacing each edge of G by two directed edges, one in each direction. We will show that if G has a Hamilton decomposition with certain additional structure, then G ↔ × C ↔ n × K ↔ 2 has a directed Hamilton decomposition. In particular, it will follow that the bidirected cubes Q ↔ 2 m + 1 for m ⩾ 2 are decomposable into 2 m + 1 directed Hamilton cycles and that a product of cycles C ↔ n 1 × ⋯ × C ↔ n m × K ↔ 2 is decomposable into 2 m + 1 directed Hamilton cycles if n i ⩾ 3 and m ⩾ 2 .