Hamilton-Laceability of Honeycomb Toriodal graphs

In this paper, we build on the work of Alspach, Chen, and Dean [2] who showed that proving the hamiltonicity of the Cayley graph of the the dihedral group Dn reduces to showing that certain cubic, connected, bipartite graphs (called honeycomb toroidal graphs) are hamilton laceable; that is, any two vertices at odd distance from each other can be joined by a hamilton path. Alspach, Chen, and Dean were able to prove hamilton-laceability in the case where the order of a honeycomb toroidal graph is divisible by four, thus enabling them to prove that the Cayley graph of Dn is hamiltonian when n is even. In this paper, we examine the case where the order of such graphs is equal to 2 mod 4. We shall show that the problem of hamilton-laceability of such graphs reduces to the degenerate case; that is, the hamiltonlaceability of honeycomb toroidal graphs consisting of a hamilton cycle C = v0v1 · · · v2n−1 together with edges vivi+s, i = 0, 2, . . . , 2n−2 where s is a fixed odd number known as the shift. In addition, we show that when n ≥ 4s, such graphs are hamilton-laceable.