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

Charles Colbourn, Petteri Kaski, Patric R J Stergrd, David A. Pike, Olli Pottonen

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


A Hanani triple system of order 6n+1, HATS(6n+1), is a decomposition of the complete graph K6n+1 into 3n sets of 2n disjoint triangles and one set of n disjoint triangles. A nearly Kirkman triple system of order 6n, NKTS(6n), is a decomposition of K6n-F into 3n-1 sets of 2n disjoint triangles; here F is a one-factor of K6n. 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 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.

Original languageEnglish (US)
Pages (from-to)827-834
Number of pages8
JournalDiscrete Mathematics
Issue number10-11
StatePublished - Jun 6 2011


  • Hanani triple system
  • Nearly Kirkman triple system
  • Resolvable design
  • Steiner triple system

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics


Dive into the research topics of 'Nearly Kirkman triple systems of order 18 and Hanani triple systems of order 19'. Together they form a unique fingerprint.

Cite this