Laceable knights

Abstract

A bipartite graph is Hamilton-laceable if for any two vertices in different parts there is a Hamiltonian path from one to the other. Using two main ideas (an algorithm for finding Hamiltonian paths and a decomposition lemma to move from smaller cases to larger) we show that the graph of knight ’ s moves on an m × n board is Hamilton-laceable iff m ≥ 6 , n ≥ 6 , and one of m , n is even. We show how the algorithm leads to new conjectures about Hamiltonian paths for various families, such as generalized Petersen graphs, I -graphs, and cubic symmetric graphs.