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

Keranen, Melissa S.; Pastine, Adrián

doi:10.26493/1855-3974.1610.03d

Cited by 8 publications

(8 citation statements)

References 21 publications

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“…When m and n have opposite parities, less is known. The paper [10] solves this problem when m | n, v > 6n > 36m and β ≥ 3; further results for cycle lengths of opposite parities can be found in [21]. The case (m, n) = (3, 4) is completely solved [5,14,26,29].…”

Section: Introductionmentioning

confidence: 89%

The Hamilton–Waterloo Problem with even cycle lengths

Burgess

Danziger

Traetta

2019

Discrete Mathematics

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The Hamilton-Waterloo Problem HWP(v; m, n; α, β) asks for a 2factorization of the complete graph K v or K v − I, the complete graph with the edges of a 1-factor removed, into α C m -factors and β C nfactors, where 3 ≤ m < n. In the case that m and n are both even, the problem has been solved except possibly when 1 ∈ {α, β} or when α and β are both odd, in which case necessarily v ≡ 2 (mod 4). In this paper, we develop a new construction that creates factorizations with larger cycles from existing factorizations under certain conditions. This construction enables us to show that there is a solution to HWP(v; 2m, 2n; α, β) for odd α and β whenever the obvious necessary conditions hold, except possibly if β = 1; β = 3 and gcd(m, n) = 1; α = 1; or v = 2mn/ gcd(m, n). This result almost completely settles the existence problem for even cycles, other than the possible exceptions noted above.

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Section: Introductionmentioning

confidence: 89%

The Hamilton–Waterloo Problem with even cycle lengths

Burgess

Danziger

Traetta

2019

Discrete Mathematics

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“…A somewhat more restrictive result on

H W P (K_{n}; T_{1}, T_{2}; α, γ)

for uniform 2‐factor types

T_{1}

and

T_{2}

, with exactly one of them bipartite, is proved in [31, Theorem 1.2]. The same authors also have solutions to

H W P (K_{n}; T_{1}, T_{2}; α, γ)

for certain nonuniform 2‐factor types [30, Corollary 10.9], and some of these cases involve exactly one bipartite 2‐factor type.…”

Section: The Hamilton–waterloo Problem For Complete Multigraphsmentioning

confidence: 99%

“…Many solutions to

H W P (K_{n \times m}; T_{1}, T_{2}; α_{1}, α_{2})

with

T_{1}

and

T_{2}

both of uniform types were found in [5,14,16,31]. As far as we can tell, the only results for nonuniform 2‐factors are found in [30]; the authors obtained extensive solutions for the complete equipartite graphs

K_{n \times m}

with

n

odd, assuming the parameters satisfy some tedious arithmetic conditions.…”

Section: The Hamilton–waterloo Problem For Complete Equipartite Multigraphsmentioning

confidence: 99%

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 somewhat more restrictive result on HW P (K n ; T 1 , T 2 ; α, γ) for uniform 2-factor types T 1 and T 2 , with exactly one of them bipartite, is proved in [27,Theorem 1.2]. The same authors also have solutions to HW P (K n ; T 1 , T 2 ; α, γ) for certain non-uniform 2-factor types [26, Corollary 10.9], and some of these cases involve exactly one bipartite 2-factor type.…”

Section: The Hamilton-waterloo Problem For Complete Multigraphsmentioning

confidence: 99%

“…Many solutions to HW P (K n×m ; T 1 , T 2 ; α 1 , α 2 ) with T 1 and T 2 both of uniform types were found in [5,14,16,27]. As far as we can tell, the only results for non-uniform 2-factors are found in [26]; the authors obtained extensive solutions for the complete equipartite graphs K n×m with n odd, assuming the parameters satisfy some tedious arithmetic conditions.…”

Section: The Hamilton-waterloo Problem For Complete Equipartite Multi...mentioning

confidence: 99%

Resolvable Cycle Decompositions of Complete Multigraphs and Complete Equipartite Multigraphs via Layering and Detachment

Bahmanian¹,

Šajna²

2018

Preprint

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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 α-resolvable cycle decompositions.

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

Cited by 8 publications

References 21 publications

The Hamilton–Waterloo Problem with even cycle lengths

The Hamilton–Waterloo Problem with even cycle lengths

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

Resolvable Cycle Decompositions of Complete Multigraphs and Complete Equipartite Multigraphs via Layering and Detachment

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