### Mutually orthogonal cycle systems

#### Abstract

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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MANUSCRIPTDOI: https://doi.org/10.26493/1855-3974.2692.86d

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications