Hamiltonian decompositions of prisms over cubic graphs

The Prism Over the Middle-levels Graph is Hamiltonian

“…As sh t (B) is on T 1 for all t ≥ 0, then, by Lemma 7 (iii), for any E on C , there is at most one t e < 2k + 1 so that sh t e (E) is on T 2 . However, A x is on both P 1 and C , and |P 1 | ≤ |P 2 | leads to a contradiction with the previous statement as |P 1 | ≤ |P 2 | implies (Lemma 7 (iii)) that at least two distinct shifts of A x have to be on P 2 ⊂ S, hence on T 2 . The proof is complete.…”

“…By [9] (see also [1]), any 3-connected cubic graph has a hamiltonian prism. Thus, the assertion follows from Theorem 8.…”

“…For a nice survey of results on k-walks, k-trees and related topics, we refer the reader to Ellingham [4]. The prism over a graph G is the Cartesian product G2K 2 of G with the complete graph K 2 [1,7,9]. Thus, it consists of two copies of G and a 1-factor joining the corresponding vertices.…”

Hamilton cycles in prisms

“…Edges of the form vv * are refered to as vertical. In G K 2 , 2-factors induce a useful edge coloring of the graph G (a similar coloring scheme related to hamiltonian decompositions was defined in [7]). Any 2-factor F in G K 2 induces a coloring of a subset of E(G) in three colors (blue, yellow, and green), defined as follows.…”

Hamiltonian cycles in symmetric graphs.

“…meaning that (s,j, /, [1]) represents the same vertex as (s + j,j, 1 -/), which gives us the desired result. The case (s,j,f, [n]) is done similarly.…”