A note on the Hamilton–Waterloo problem with C8-factors and Cm

Abstract

In this paper, we almost completely solve the Hamilton-Waterloo problem with C 8 -factors and C m -factors where the number of vertices is a multiple of 8m.

Infinitely many cyclic solutions to the Hamilton–Waterloo problem with odd length cycles

“…. , δ 8 ) = (30,31,32,33,35,36,37,34); also each δ h lies in [1,45]. Therefore, conditions 1, 2, and 3 of the above definition are satisfied, and hence B is alternating.…”

“…Note that ϕ m (−x) = −ϕ m (x) for any x ∈ Z − 91 \ (D ∪ −D). In particular, for m = 6 the map ϕ m acts on [2,44] \ D as follows: −1), (4, 1), (6, −1), (7, 1), (8, 1), (9, 1), (10, −1), (11, 1), (12, −1), (13, 1), (14, −1), (15, −1), (16,1), (17, −2), (18, −1), (19, −1), (20, −1), (21, 1), (22, −1), (23, 1), (24, −2), (25, −1), (26,2), (27, −1), (28,1), (29,1), (30, −2), (31, 1), (32,2), (33,7), (34, −1), (35, 1), (36, −1), (37, 1), (38, −1), (39, 8), (40, 1), (41, 1), (42, −1), (43, 1), (44, −1)}.…”

“…11,12,13,15,16,17,14) for i = 2,(20,21,22,23,25,26,27,24) for i = 3,(30,31,32,33, 35, 36, 37,34) for i = 4.It follows that the cycles B 2 , B 3 , B 4 are alternating. Also, ∆B 2 = ±[10,17], ∆B 3 = ±[20,27], and ∆B 4 = ±[30, 37].…”

“…Surprisingly, very little is known when π or π ′ contain odd terms, even when all terms of π and all terms of π ′ coincide. For example, HWP(v; [4 v/4 ], [ℓ v/ℓ ]; r, r ′ ) has been dealt with in [28,29] and completely solved only when ℓ = 3 in [5,17,33], while HWP(v; [3 v/3 ], [v]; ⌊v/2⌋ − 1, 1) is still open (see, [19,20,25]). Other results can be found in [3,13].…”

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, … , }.…”