Steiner pentagon systems (Q800362)

A Steiner pentagon system is a partition of the complete graph \(K_ n\) into pentagons such that each pair of distinct vertices is joined by a path of length two in exactly one of the pentagons. Such systems give rise to quasigroups (in a manner similar to the construction of Steiner quasigroups from Steiner triple systems). The authors show that a Steiner pentagon system exists precisely for all \(n\equiv 1\) or 5 mod 10, with the exception \(n=15\). Their construction uses perpendicular arrays and self-orthogonal latin squares with symmetric mate.