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

A Hanani triple system of order 6n+16n+1, HATS(6n+1), is a decomposition of the complete graph K6n+1K6n+1 into 3n3n sets of 2n2n disjoint triangles and one set of nn disjoint triangles. A nearly Kirkman triple system of order 6n6n, NKTS(6n), is a decomposition of K6n−FK6n−F into 3n−13n−1 sets of 2n2n disjoint triangles; here FF is a one-factor of K6nK6n. The Hanani triple systems of order 6n+16n+1 and the nearly Kirkman triple systems of order 6n6n can be classified using the classification of the Steiner triple systems of order 6n+16n+1. This is carried out here for n=3n=3: There are 3787983639 isomorphism classes of HATS(19)s and 25328 isomorphism classes of NKTS(18)s. Several properties of the classified systems are tabulated. In particular, seven of the NKTS(18)s have orthogonal resolutions, and five of the HATS(19)s admit a pair of resolutions in which the almost parallel classes are orthogonal.