Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Mutually orthogonal cycle systems   An ${\ell}$-cycle system ${\mathcal F}$ of a graph $\Gamma$ is a set of
${\ell}$-cycles which partition the edge set of $\Gamma$. Two such cycle
systems ${\mathcal F}$ and ${\mathcal F}'$ are said to be {\em orthogonal} if
no two distinct cycles from ${\mathcal F}\cup {\mathcal F}'$ 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 $\ell$-cycle systems of $\Gamma$ is said to be a set of
mutually orthogonal cycle systems of $\Gamma$.
  Let $\mu(\ell,n)$ (respectively, $\mu'(\ell,n)$) be the maximum integer $\mu$
such that there exists a set of $\mu$ mutually orthogonal (cyclic) $\ell$-cycle
systems of the complete graph $K_n$. We show that if $\ell\geq 4$ is even and
$n\equiv 1\pmod{2\ell}$, then $\mu'(\ell,n)$, and hence $\mu(\ell,n)$, is
bounded below by a constant multiple of $n/\ell^2$. In contrast, we obtain the
following upper bounds: $\mu(\ell,n)\leq n-2$; $\mu(\ell,n)\leq
(n-2)(n-3)/(2(\ell-3))$ when $\ell \geq 4$; $\mu(\ell,n)\leq 1$ when
$\ell>n/\sqrt{2}$; and $\mu'(\ell,n)\leq n-3$ when $n \geq 4$. We also obtain
computational results for small values of $n$ and $\ell$.

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The categories of this paper are: atom-ph
26
The categories of this paper are: bayes-an
19
The categories of this paper are: eess
21
The categories of this paper are: adap-org
29
The categories of this paper are: funct-an
4
The categories of this paper are: cond-mat
10
The categories of this paper are: q-bio
11
The categories of this paper are: hep-lat
8
The categories of this paper are: hep-ex
16
The categories of this paper are: stat
36
The categories of this paper are: cmp-lg
15
The categories of this paper are: nucl-ex
20
The categories of this paper are: acc-phys
1
The categories of this paper are: cs
3
The categories of this paper are: physics
22
The categories of this paper are: chao-dyn