The Hamilton–Waterloo problem: the case of Hamilton cycles and triangle-factors

Abstract

We discuss a special case of the Hamilton-Waterloo problem in which a 2-factorization of Kn is sought consisting of 2-factors of two kinds: Hamiltonian cycles, and triangle-factors. We determine completely the spectrum of solutions for several inÿnite classes of orders n.

The Hamilton—Waterloo problem: The case of triangle‐factors and one Hamilton cycle

“…In fact this paper is completely dedicated to the case of the Hamilton-Waterloo problem with triangle-factors and exactly one Hamilton cycle. As noted in [9], this is the most difficult case of this problem. We will base our work on several recursive constructions as well as direct constructions for some small orders.…”

“…So since s + r = n−1 2 it is then convenient to just give the number of Hamilton cycles r in the HW(r, s; n, 3) when determining the spectrum for this problem. The following two theorems are proven in [9]. Theorem 1.2 [9] (a) Let n = 6k + 3 and assume that k ≡ 1 (mod 3), then there is a solution to the Hamilton-Waterloo problem HW(r, s; n, 3) with triangle-factors and exactly r Hamilton cycles for every 0 ≤ r ≤ n−1 2 , except possibly when r = 1.…”

“…The following two theorems are proven in [9]. Theorem 1.2 [9] (a) Let n = 6k + 3 and assume that k ≡ 1 (mod 3), then there is a solution to the Hamilton-Waterloo problem HW(r, s; n, 3) with triangle-factors and exactly r Hamilton cycles for every 0 ≤ r ≤ n−1 2 , except possibly when r = 1. (b) Let n = 6k + 3 and assume that k ≡ 0, 2 (mod 3), then there is a solution to the Hamilton-Waterloo problem HW(r, s; n, 3) with triangle-factors and exactly r Hamilton cycles for every n+3 6 ≤ r ≤ n−1 2 , except possibly when r = n+3 6 + 1.…”

“…Finally, one can check that pairs of elements of the form (x 0 , (x + 10) 1 ), (y 0 , (y + 8) 2 ) and (z 1 , (z + 6) 2 ) (for x, y, z ∈ Z 17 ) have yet to appear together in a block. By Lemma 1 of [9] these sets of edges can be ordered to form a Hamilton cycle in V . Then C 0 is a parallel class of triples which when developed by adding all the elements of {0, 1, 2, .…”

“…In the paper by Horak, Nedela and Rosa [9] the authors consider the case of finding a 2-factorization of K n consisting of Hamilton cycles (at Hamilton) and of trianglefactors (at Waterloo), so they are considering the case of HW(r, s; n, 3). In this case the necessary condition is just that n ≡ 3 (mod 6), when s ≥ 1.…”