Cyclic cycle systems of the complete multipartite graph

An ℓ-cycle system F of a graph Γ is a set of ℓ-cycles which partition the edge set of Γ. Two such cycle systems F and F ′ are said to be orthogonal if no two distinct cycles from F ∪ F ′ share more than one edge. Orthogonal cycle systems naturally arise from face 2-colourable polyehdra and in higher genus from Heffter arrays with certain orderings. A set of pairwise orthogonal ℓ-cycle systems of Γ is said to be a set of mutually orthogonal cycle systems of Γ.Let µ(ℓ, n) (respectively, µ ′ (ℓ, n)) be the maximum integer µ such that there exists a set of µ mutually orthogonal (cyclic) ℓ-cycle systems of the complete graph K n . We show that if ℓ ≥ 4 is even and n ≡ 1 (mod 2ℓ), then µ ′ (ℓ, n), and hence µ(ℓ, n), is bounded below by a constant multiple of n/ℓ 2 . In contrast, we obtain the following upper bounds:√ 2; and µ ′ (ℓ, n) ≤ n − 3 when n ≥ 4. We also obtain computational results for small values of n and ℓ.

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“…The idea is to construct cyclic systems using so-called balanced sets of differences. The following definitions and lemma appear in [10].…”

Section: Preliminary Lemmas For Cycle Length Greater Thanmentioning

confidence: 99%

Mutually orthogonal cycle systems

2022

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“…Then the difference associated with an edge {a, b} is defined to be the minimum value in The idea is to construct cyclic systems using so-called balanced sets of differences. The following definitions and lemma appear in [10].…”

Section: Preliminary Lemmas For Cycle Length Greater Thanmentioning

confidence: 99%

Mutually orthogonal cycle systems

Burgess¹,

Cavenagh²,

Pike³

2022

Preprint

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An ℓ-cycle system F of a graph Γ is a set of ℓ-cycles which partition the edge set of Γ. Two such cycle systems F and F ′ are said to be orthogonal if no two distinct cycles from F ∪ F ′ share more than one edge. Orthogonal cycle systems naturally arise from face 2-colourable polyehdra and in higher genus from Heffter arrays with certain orderings. A set of pairwise orthogonal ℓ-cycle systems of Γ is said to be a set of mutually orthogonal cycle systems of Γ.Let µ(ℓ, n) (respectively, µ ′ (ℓ, n)) be the maximum integer µ such that there exists a set of µ mutually orthogonal (cyclic) ℓ-cycle systems of the complete graph K n . We show that if ℓ 4 is even and n ≡ 1 (mod 2ℓ), then µ ′ (ℓ, n), and hence µ(ℓ, n), is bounded below by a constant multiple of n/ℓ 2 . In contrast, we obtain the following upper bounds: µ(ℓ, n) n−2; µ(ℓ, n) (n−2)(n−3)/(2(ℓ−3)) when ℓ 4; µ(ℓ, n) 1 when ℓ > n/ √ 2; and µ ′ (ℓ, n) n−3 when n 4. We also obtain computational results for small values of n and ℓ.

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“…Results about regular cycle decompositions of the complete multipartite graph via relative difference families can be found in [9,11,20,22]. Now, in order to present the connection between relative Heffter arrays and relative difference families, we have to introduce the concept of simple ordering.…”

Section: Relation With Relative Difference Families and Decomposition...mentioning

confidence: 99%

A generalization of Heffter arrays

Costa

Morini

Pasotti

et al. 2019

J of Combinatorial Designs

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In this paper we define a new class of partially filled arrays, called relative Heffter arrays, that are a generalization of the Heffter arrays introduced by Archdeacon in 2015. Let v = 2nk + t be a positive integer, where t divides 2nk, and let J be the subgroup of Zv of order t. A Ht(m, n; s, k) Heffter array over Zv relative to J is an m × n partially filled array with elements in Zv such that: (i) each row contains s filled cells and each column contains k filled cells; (ii) for every x ∈ Z 2nk+t \ J, either x or −x appears in the array; (iii) the elements in every row and column sum to 0. In particular, here we study the existence for t = k of integer (i.e. the entries are chosen in ± 1, . . . , 2nk+t 2 and the sums are zero in Z) square relative Heffter arrays.

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Cyclic cycle systems of the complete multipartite graph

Abstract: In this paper, we study the existence problem for cyclic ℓ-cycle decompositions of the graph Km[n], the complete multipartite graph with m parts of size n, and give necessary and sufficient conditions for their existence in the case that 2ℓ |

Cited by 5 publications

References 33 publications

Mutually orthogonal cycle systems

Mutually orthogonal cycle systems

Mutually orthogonal cycle systems

A generalization of Heffter arrays

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