A complete solution to the two-table Oberwolfach problems

We prove that K n + I, the complete graph of even order with a 1‐factor duplicated, admits a decomposition into 2‐factors, each a disjoint union of cycles of length m ≥ 5 if and only if m ∣ n, except possibly when m is odd and n = 4 m. In addition, we show that K n + I admits a decomposition into 2‐factors, each a disjoint union of cycles of lengths m 1 , … , m t, whenever m 1 , … , m t are all even.

m_{1}, \dots, m_{t}

all even , for

t = 2

, and for

n \leq 40

. In general, however, it is still open.…”

Section: Introductionmentioning

confidence: 90%

On the spouse‐loving variant of the Oberwolfach problem

Bolohan

Buchanan

Burgess

et al. 2019

“…Proof. The existence of a path P ∈ P(0, γ ; γ + 1, δ) whose end-vertices u i and v i satisfy the assertion is proven in [1] by using the terminology of α-labelings (see, also, [44]).…”

Section: Proofmentioning

confidence: 99%

A Complete Solution to the Existence of ‐Cycle Frames of Type

Buratti

Cao

Dai

et al. 2016

“…In the nonuniform case, it is known that OP( F ) has no solution if

F \in {false[3^{2} false], false[3^{4} false], false[4, 5 false], false[3^{2}, 5 false]}

, but that there is a solution for every other instance in which

v \leq 40

. The problem OP( F ) has been solved if F has exactly two components (see also for 2‐factorizations of the complete multipartite graph), and if F is bipartite . Although complete solutions have been given for infinite families of orders , the general problem is still open.…”

Section: Introductionmentioning

confidence: 99%

On the Hamilton–Waterloo problem with odd cycle lengths

Burgess

Danziger

Traetta

2017

Let Kv∗ denote the complete graph Kv if v is odd and Kv−I, the complete graph with the edges of a 1‐factor removed, if v is even. Given nonnegative integers v,M,N,α,β, the Hamilton–Waterloo problem asks for a 2‐factorization of Kv∗ into α CM‐factors and β CN‐factors, with a Cℓ‐factor of Kv∗ being a spanning 2‐regular subgraph whose components are ℓ‐cycles. Clearly, M,N≥3, M∣v, N∣v and α+β=⌊v−12⌋ are necessary conditions. In this paper, we extend a previous result by the same authors and show that for any odd v≠MNgcd(M,N) the above necessary conditions are sufficient, except possibly when α=1, or when β∈{1,3}. Note that in the case where v is odd, M and N must be odd. If M and N are odd but v is even, we also show sufficiency but with further possible exceptions. In addition, we provide results on 2‐factorizations of the complete equipartite graph and the lexicographic product of a cycle with the empty graph.