A Generalization of the Hamilton–Waterloo Problem on Complete Equipartite Graphs

Keranen, Melissa S.; Pastine, Adrián

doi:10.1002/jcd.21560

Cited by 10 publications

(22 citation statements)

References 15 publications

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“…Therefore, improving such results would allow us to remove some of the possible exceptions in Theorems and . We note that some of the cases arising from exception 2 of Theorem have been solved in when

gcd (M, N, t) > 1

.…”

Section: Resultsmentioning

confidence: 99%

“…In this section, we first solve

HWP (C_{M} false[n false]; M, M n; α, β)

for odd

M, n \geq 3

, except possibly when

β = 1

. We point out that the same result has been independently proven with different techniques in . Theorem Let

M, n \geq 3

be odd integers and

N = M n

.…”

Section: Factorizing Cmfalse[nfalse] and Cnfalse[mfalse] Into Cm‐factmentioning

confidence: 99%

“…For instance, the case ( , ) = (3, 3 ) is considered in [5], where the authors solve HWP( ; 3, 3 ; , ) when is odd and ≠ 1, and give a partial solution for even . The Hamilton-Waterloo problem for complete equipartite graphs is considered in [30].…”

Section: Introductionmentioning

confidence: 99%

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On the Hamilton–Waterloo problem with odd cycle lengths

Burgess

Danziger

Traetta

2017

J of Combinatorial Designs

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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.

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gcd (M, N, t) > 1

.…”

Section: Resultsmentioning

confidence: 99%

“…In this section, we first solve

HWP (C_{M} false[n false]; M, M n; α, β)

for odd

M, n \geq 3

, except possibly when

β = 1

. We point out that the same result has been independently proven with different techniques in . Theorem Let

M, n \geq 3

be odd integers and

N = M n

.…”

Section: Factorizing Cmfalse[nfalse] and Cnfalse[mfalse] Into Cm‐factmentioning

confidence: 99%

See 1 more Smart Citation

On the Hamilton–Waterloo problem with odd cycle lengths

Burgess

Danziger

Traetta

2017

J of Combinatorial Designs

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“…Note that a result similar to Lemma 3.2 has been proven in [18] in the case where M ≥ 3 is odd and n > 1. Proof.…”

Section: Skolem Sequencesmentioning

confidence: 59%

“…Note that the above Lemma has been independently proven in [18] with different techniques. An alternative proof in the case where M is odd can be found in [12].…”

Section: Skolem Sequencesmentioning

confidence: 99%

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

Burgess

Danziger

Traetta

2018

Discrete Mathematics

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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.

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Resolvable cycle decompositions of complete multigraphs and complete equipartite multigraphs via layering and detachment

Bahmanian

Šajna

2021

J of Combinatorial Designs

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We construct new resolvable decompositions of complete multigraphs and complete equipartite multigraphs into cycles of variable lengths (and a perfect matching if the vertex degrees are odd). We develop two techniques: layering, which allows us to obtain 2‐factorizations of complete multigraphs from existing 2‐factorizations of complete graphs, and detachment, which allows us to construct resolvable cycle decompositions of complete equipartite multigraphs from existing resolvable cycle decompositions of complete multigraphs. These techniques are applied to obtain new 2‐factorizations of a specified type for both complete multigraphs and complete equipartite multigraphs, with the emphasis on new solutions to the Oberwolfach Problem and the Hamilton–Waterloo Problem. In addition, we show existence of some thickly resolvable cycle decompositions.

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A Generalization of the Hamilton–Waterloo Problem on Complete Equipartite Graphs

Cited by 10 publications

References 15 publications

On the Hamilton–Waterloo problem with odd cycle lengths

On the Hamilton–Waterloo problem with odd cycle lengths

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

Resolvable cycle decompositions of complete multigraphs and complete equipartite multigraphs via layering and detachment

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