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

Abstract

A Hanani triple system of order 6 n + 1 , HATS ( 6 n + 1 ) , is a decomposition of the complete graph K 6 n + 1 into 3 n sets of 2 n disjoint triangles and one set of n disjoint triangles. A nearly Kirkman triple system of order 6 n , NKTS ( 6 n ) , is a decomposition of K 6 n − F into 3 n − 1 sets of 2 n disjoint triangles; here F is a one-factor of K 6 n . The Hanani triple systems of order 6 n + 1 and the nearly Kirkman triple systems of order 6 n can be classified using the classification of the Steiner triple systems of order 6 n + 1 . This is carried out here for n = 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.