The Hamilton‐Waterloo problem for Hamilton cycles and triangle‐factors

Abstract

Abstract:In this article, we consider the Hamilton-Waterloo problem for the case of Hamilton cycles and triangle-factors when the order of the complete graph K n is even. We completely solved the problem for the case n ≡ 24 (mod 36). For the cases n ≡ 0 (mod 18) and n ≡ 6 (mod 36), we gave an almost complete solution. q 2011 Wiley Periodicals, Inc. J Combin Designs

The Hamilton-Waterloo Problem for C3-Factors and Cn-Factors

“…In this section, we shall solve the three left cases in [22] for the existence of an HW(v; 3, 7, α, β) when v ≡ 21 (mod 42). Then we continue to consider the case v ≡ 0 (mod 42).…”

“…Proof: Let the vertex set be Z 21 × Z 4 , and the four parts of K 4 [21] be Z 21 × {i}, i ∈ Z 4 . For any α > 0, the required α C 3 -factors are…”

“…So when H = K v (for v odd) or K v minus a 1-factor (for v even), (24,5,6), (24,9,2), (48,17,6). (21,4,6), (21,6,4).…”

A note on the Hamilton–Waterloo problem with C8-factors and CmWang1, Cao2 2018Discrete Mathematics90100View full textAdd to dashboardCite

“…For the case (m, n) ∈ {(3, 15), (5,15), (4,6), (4,8), (4,16), (8,16)}, see [1]. The existence of an HW(v; 4, n; α, β) for odd n ≥ 3 has been solved except possibly when v = 8n and α = 2, see [13,21,24].…”

“…Many infinite classes of HW(v; 3, 3x; α, β)s are constructed in [3]. For more results on the Hamilton-Waterloo problem, the reader can see [5,8,11,15,16,25]. In this paper, we focus on the existence of an HW(8mt; 8, m; α, β).…”