On the Hamilton–Waterloo Problem with cycle lengths of distinct parities

Abstract

Let K * v denote the complete graph K v if v is odd and K v − I, the complete graph with the edges of a 1-factor removed, if v is even. Given non-negative integers v, M, N, α, β, the Hamilton-Waterloo problem asks for a 2-factorization of K * v into α C M -factors and β C N -factors. Clearly, M, N ≥ 3, M | v, N | v and α + β = ⌊ v−1 2 ⌋ are necessary conditions.Very little is known on the case where M and N have different parities. In this paper, we make some progress on this case by showing, among other things, that the above necessary conditions are sufficient whenever M |N , v > 6N > 36M , and β ≥ 3.

Completing the spectrum of almost resolvable cycle systems with odd cycle length

“…k = 17: C8 = (∞, (0, 0), (7, 2), (7, 1), (6, 1), (−8, 1), (−6, 1), (5, 1), (−5, 1), (8, 1), (8,3), (−8, 2), (6, 2), (−2, 2), (−7, 2), (4, 2), (8,2)). k = 33: C8 = (∞, (0, 0), (15,2), (15, 1), (9, 1), (−9, 1), (10, 1), (−10, 1), (11, 1), (−11, 1), (12, 1), (13, 1), (−12, 1), (14, 1), (−14, 1), (−16, 1), (−13, 1), (16,1), (16,3), (−16, 2), (8,2), (12,2), (−14, 2), (16,2), (−11, 2), (14, 2), (−9, 2), (10, 2), (−10, 2), (11,2), (−4, 2), (13,2), (−15, 2)). k = 41: C8 = (∞, (0, 0), (19,2), (19,1), (−12, 1), (13, 1), (−13, 1), (11, 1), (−11, 1), (12, 1), (−15, 1), (14, 1), (−14, 1), (18, 1), (15, 1), (16, 1), (−17, 1), (17, 1), (−18, 1), (−20, 1), (−16, 1), (20,1), (20,3), (−20, 2), (10, 2), (13,2), (−13, 2), (14,2), (−11, 2), (12,2), (−12, 2), (17,2), (−5, 2), (16,2), (−19, 2), (18,2), (−14, 2), (20,2), (−16, 2), (15,2), (−18, 2)).…”

“…k = 27: (a2, b2) = ( k+1 2 , 2), d2 = 2. C7 = ((0, 2), (−13, 1), (−11, 2), (0, 0), (1, 2), (2, 0), (4, 2), (1, 0), (5,2), (−7, 0), (2, 2), (4, 0), (9, 2), (−6, 0), (7, 2), (−4, 0), (3,2), (−5, 0), (13,2), (3, 0), (−5, 2), (−2, 0), (−8, 2), (−3, 0), (−7, 2), (6, 0), (−4, 2)); C8 = (∞, (5, 0), (−2, 2), (13, 1), (11, 1), (−12, 1), (−11, 1), (−8, 1), (9, 1), (−9, 1), (12, 1), (−10, 1), (10, 1), (−6, 1), (7, 1), (7, 3), (8, 2), (11,2), (−6, 2), (6, 2), (12, 2), (−10, 2), (10, 2), (−3, 2), (−12, 2), (−1, 2), (−9, 2)). k = 35: (a2, b2) = ( k+1 2 , 2), d2 = 2.…”

“…k = 35: (a2, b2) = ( k+1 2 , 2), d2 = 2. C7 = ((1, 2), (−15, 1), (2, 2), (−2, 0), (3, 2), (−3, 0), (4, 2), (−5, 0), (5,2), (−6, 0), (6, 2), (−7, 0), (7, 2), (−8, 0), (8,2), (−9, 0), (11,2), (8, 0), (9, 2), (7, 0), (−9, 2), (5, 0), (−8, 2), (4, 0), (−7, 2), (3, 0), (−6, 2), (−4, 0), (−5, 2), (0, 0), (−4, 2), (2, 0), (−1, 2), (6, 0), (−2, 2)); C8 = (∞, (1, 0), (−16, 2), (17, 1), (−16, 1), (16, 1), (−12, 1), (−17, 1), (−13, 1), (13, 1), (−14, 1), (15, 1), (14, 1), (−10, 1), (11, 1), (−11, 1), (12, 1), (−8, 1), (9, 1), (9, 3), (10, 2), (−12, 2), (12, 2), (−11, 2), (14,2), (−13, 2), (16,2), (−10, 2), (−15, 2), (13,2), (−3, 2), (15,2), (0, 2), (−14, 2), (17,2)).…”

“…For k = 39, , the cycle C 8 is listed as below. C8 = (∞, (0, 0), (−19, 2), (18, 1), (−18, 1), (15, 1), (−19, 1), (19, 1), (−16, 1), (−14, 1), (16, 1), (−15, 1), (17, 1), (−12, 1), (12, 1), (−11, 1), (14, 1), (−13, 1), (13, 1), (−9, 1), (10, 1), (10, 3), (11,2), (−14, 2), (13, 2), (−13, 2), (15,2), (−15, 2), (14,2), (−5, 2), (19,2), (16,2), (−16, 2), (17,2), (−1, 2), (−17, 2), (−12, 2), (−10, 2), (−18, 2)).…”

Further results on almost resolvable cycle systems and the Hamilton–Waterloo problem

“…Burgess et al [13] have made significant progress on the existence of an HW( ; , , , ) in which and are all odd as we can see in the theorem below. We point the reader towards [14,15,21,46] for more results on the HamiltonWaterloo problem. For brevity, we use [ , ] to denote the integer set { , + 1, + 2, … , }.…”