Nearly Kirkman triple systems of order 18 and Hanani triple systems of order 19

Abstract

A Hanani triple system of order $6n+1$, $\mathrm{HATS}(6n+1)$, is a decomposition of the complete graph $K_{6n+1}$ into $3n$ sets of $2n$ disjoint triangles and one set of $n$ disjoint triangles. A nearly Kirkman triple system of order $6n$, $\mathrm{NKTS}\left(6n\right)$, is a decomposition of $K_{6n}-F$ into $3n-1$ sets of $2n$ disjoint triangles; here $F$ is a one-factor of $K_{6n}$. The Hanani triple systems of order $6n+1$ and the nearly Kirkman triple systems of order $6n$ can be classified using the classification of the Steiner triple systems of order $6n+1$. This is carried out here for $n=3$: There are 3787983639 isomorphism classes of $\mathrm{HATS}\left(19\right)$s and 25328 isomorphism classes of $\mathrm{NKTS}\left(18\right)$s. Several properties of the classified systems are tabulated. In particular, seven of the $\mathrm{NKTS}\left(18\right)$s have orthogonal resolutions, and five of the $\mathrm{HATS}\left(19\right)$s admit a pair of resolutions in which the almost parallel classes are orthogonal.