Mutually orthogonal cycle systems - Burgess, Andrea (Ars mathematica contemporanea, 2023)
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 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) v$\ell$?-cycle systems of the complete graph ?$K_n$?. We show that if ?$\ell \ge 4$? is even and v$n \equiv 1 \pmod 2\ell$?, 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: ?$\mu(\ell,n) \le n-2$?; ?$\mu(\ell,n) \le (n-2)(n-3)/(2(\ell-3))$? when ?$\ell \ge 4$?; ?$\mu(\ell, n) \le 1$? when ?$\ell > n/\sqrt{2}$?; and ?$\mu'(\ell,n) \le n-3$? when ?$n \ge 4$?. We also obtain computational results for small values of ?$n$? and ?$\ell$?.
Keywords: ciklični sistemi ciklov; completely-reducible; cyclic cycle systems; Heffter arrays; Heffterjeva polja; orthogonal cycle decompositions; popolnoma reducibilen; pravokotne ciklične dekompozicije; superenostaven; super-simple