On the generalized Oberwolfach problem

Abstract

The generalized Oberwolfach problem OP t (2w + 1; N 1 , N 2 ,. .. , N t ; α 1 , α 2 ,. .. , α t) asks for a factorization of K 2w+1 into α i C Ni-factors (where a C Ni-factor of K 2w+1 is a spanning subgraph whose components are cycles of length N i ≥ 3) for i = 1, 2,. .. , t. Necessarily, N = lcm(N 1 , N 2 ,. .. , N t) is a divisor of 2w + 1 and w = t i=1 α i. For t = 1 we have the classic Oberwolfach problem. For t = 2 this is the well-studied Hamilton-Waterloo problem, whereas for t ≥ 3 very little is known. In this paper, we show, among other things, that the above necessary conditions are sufficient whenever 2w + 1 ≥ (t + 1)N , α i > 1 for every i ∈ {1, 2,. .. , t}, and gcd(N 1 , N 2 ,. .. , N t) > 1. We also provide sufficient conditions for the solvability of the generalized Oberwolfach problem over an arbitrary graph and, in particular, the complete equipartite graph.

On the Hamilton-Waterloo problem: the case of two cycles sizes of different parity

“…In this paper, we consider the case of m and n being of different parity. This case has gained attention recently, where it has been shown that the necessary conditions are sufficient for a (m, n)− HWP(v; r, s) whenever m | n, v > 6n > 36m, and s ≥ 3 [10]. We provide a complementary result to this in our main theorem, which covers cases in which m ∤ n and solves a major portion of the problem.…”

Gregarious Decompositions of Complete Equipartite Graphs and Related Structures

“…Still concerning cycles, there are many publications dealing with decompositions in which more than one cycle length occurs. In a very recent survey [24], Burgess, Danziger, and Traetta summarized the results of that kind and listed 62 reference items. In particular, Section 2.1.2 of that manuscript described a method based on the so-called "row-sum matrices", by which resolvable gregarious decompositions of K n * m and C n * m can be generated.…”

The Hamilton–Waterloo Problem with even cycle lengths

“…Let n > m ≥ 3 be odd integers with m ∤ n, and let g = gcd(m, n). We first prove a result that is a consequence of Theorem 2.5 and the following theorem, which is a special case of Theorem 1.4 of [13], taking t = 1, 2. We now apply Theorem 2.5 to obtain the following result.…”

“…[13]). Let 1 ≤ m ′ < n ′ ≤ N be odd integers such that m ′ and n ′ are divisors of N. Then HWP(C g[N]; gm ′ , gn ′ ; α, β) has a solution whenever g ≥ 3, α + β = N and α, β = 1.…”