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

doi:10.1016/j.disc.2011.02.005

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 journal › Article › peer-review

8 Scopus citations

Abstract

A Hanani triple system of order 6n+1, 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, NKTS(6n), 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 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 language	English (US)
Pages (from-to)	827-834
Number of pages	8
Journal	Discrete Mathematics
Volume	311
Issue number	10-11
DOIs	https://doi.org/10.1016/j.disc.2011.02.005
State	Published - Jun 6 2011

Keywords

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

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics

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10.1016/j.disc.2011.02.005

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title = "Nearly Kirkman triple systems of order 18 and Hanani triple systems of order 19",

abstract = "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.",

keywords = "Hanani triple system, Nearly Kirkman triple system, Resolvable design, Steiner triple system",

author = "Charles Colbourn and Petteri Kaski and Stergrd, \{Patric R J\} and Pike, \{David A.\} and Olli Pottonen",

note = "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 .",

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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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JO - Discrete Mathematics

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ER -

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

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