TY - JOUR
T1 - Nearly Kirkman triple systems of order 18 and Hanani triple systems of order 19
AU - Colbourn, Charles
AU - Kaski, Petteri
AU - Stergrd, Patric R J
AU - Pike, David A.
AU - Pottonen, Olli
N1 - Funding Information:
The first author was supported in part by DOD grant N00014-08-1-1070 . The second author was supported by the Academy of Finland , Grant No. 117499 . The third author was supported in part by the Academy of Finland , Grants No. 110196 , 130142 , 132122 . The fourth author was supported in part by CFI , IRIF , and NSERC . The fifth author was supported by the Graduate School in Electronics, Telecommunication and Automation, by the Nokia Foundation , and by the Academy of Finland , Grant No. 110196 .
PY - 2011/6/6
Y1 - 2011/6/6
N2 - 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.
AB - 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.
KW - Hanani triple system
KW - Nearly Kirkman triple system
KW - Resolvable design
KW - Steiner triple system
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U2 - 10.1016/j.disc.2011.02.005
DO - 10.1016/j.disc.2011.02.005
M3 - Article
AN - SCOPUS:79951943756
SN - 0012-365X
VL - 311
SP - 827
EP - 834
JO - Discrete Mathematics
JF - Discrete Mathematics
IS - 10-11
ER -