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

Buratti, Marco; Cao, Haitao; Dai, Dao‐Qing; Traetta, Tommaso

doi:10.1002/jcd.21523

Cited by 11 publications

(14 citation statements)

References 37 publications

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“…Note that Buratti et al. have proved that there exists a k ‐cycle frame of type

{false(2 k false)}^{t}

for all

t \geq 3

when k is even, and

t \geq 4

when k is odd. So for any even integer

k \geq 4

, we may use the second recursive construction to obtain a k ‐ARCS

(2 k t + 1)

for all

t \geq 3

if there is a k ‐ARCS

(2 k + 1)

.…”

Section: Resultsmentioning

confidence: 99%

“…Case 2.1: ≡ 6 (mod 24) and ≥ 30. 3 = (( − 2, 1), (2, 1), ( , 1), (3, 1), ( − 4, 1), (1, 1), … , ( − 2 − 3 , 1), (2 + 3 , 1),( − 3 , 1), (3 + 3 , 1), ( − 4 − 3 , 1), (1 + 3 , 1), … , ( +9 2 , 1), ( −9 2 , 1), ( +13 2 , 1), ( −7 2 , 1), ( +5 2 , 1), ( −11 2 , 1)), 0 ≤ ≤ −13 6 ; 4 = (( +3 2 , 1), ( +1 2 , 1), ( −5 2 , 1), ( −1 2 , 1), ( +7 2 , 1), ( −3 2 , 1), ( +1 2 , 2), ( +3 2 , 2)); 5 = (( −3 2 , 2), ( +5 2 , 2), … , ( −3 2 − , 2), ( +5 2 + , 2), … , (1, 2), ( , 2)), 0 ≤ ≤ −5 2 ; 6 = ((−( − 1), 3), (1, 3)); 7…”

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Almost resolvable k‐cycle systems with

Wang

Cao

2018

J of Combinatorial Designs

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“…Note that Buratti et al. have proved that there exists a k ‐cycle frame of type

{false(2 k false)}^{t}

for all

t \geq 3

when k is even, and

t \geq 4

when k is odd. So for any even integer

k \geq 4

, we may use the second recursive construction to obtain a k ‐ARCS

(2 k t + 1)

for all

t \geq 3

if there is a k ‐ARCS

(2 k + 1)

.…”

Section: Resultsmentioning

confidence: 99%

unclassified

Almost resolvable k‐cycle systems with

Wang

Cao

2018

J of Combinatorial Designs

Self Cite

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“…Note that an

(ℓ, λ)

‐cycle frame of type

m^{n}

is equivalent to a holey 2‐factorization of

λ K_{n \times m}

of type

[{[ℓ^{*}]}^{*}]

. The existence problem for uniform cycle frames and

({3, 5}, λ)

‐cycle frames of type

m^{n}

were completely settled in [13] and [35], respectively.…”

Section: Final Remarksmentioning

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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“…Muthusamy and Shanmuga Vadivu [32] proved the existence of a C 2kdecomposition of K m • K n . Very recently, irrespective of the parity of k, the authors of [15] actually solve the existence problem for a C k -decomposition of (K m • K n )(λ) whose cycle-set can be partitioned into 2-regular graphs containing all the vertices except those belonging to one part. A C α 4 , C β 5 -decomposition of K m • K n was given by Fu [22].…”

Section: And the Directed Hamilton Cycle Decompositions Of The Symmetmentioning

confidence: 99%

Decomposition of the tensor product of complete graphs into cycles of lengths 3 and 6

Paulraja¹,

Srimathi²

2021

Discuss. Math. Graph Theory

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By a C α 3 , C β 6-decomposition of a graph G, we mean a partition of the edge set of G into α cycles of length 3 and β cycles of length 6. In this paper, necessary and sufficient conditions for the existence of a C α 3 , C β 6decomposition of (K m × K n)(λ), where × denotes the tensor product of graphs and λ is the multiplicity of the edges, is obtained. In fact, we prove that for λ ≥ 1, m, n ≥ 3 and (m, n) = (3, 3), a C α 3 , C β 6-decomposition of (K m × K n)(λ) exists if and only if λ(m − 1)(n − 1) ≡ 0 (mod 2) and 3α + 6β = λm(m−1)n(n−1) 2 .

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

Abstract: The existence problem of a (k, λ)-cycle frame of type g u is now solved for any quadruple (k, λ, g, u).

Cited by 11 publications

References 37 publications

Almost resolvable k‐cycle systems with

Almost resolvable k‐cycle systems with

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

Decomposition of the tensor product of complete graphs into cycles of lengths 3 and 6

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