TY - JOUR
T1 - Spanning sets and scattering sets in Steiner triple systems
AU - Colbourn, Charles J.
AU - Dinitz, Jeffrey H.
AU - Stinson, Douglas R.
N1 - Funding Information:
The research was begun while the authors were attending the Design Theory Conference at Auburn University, and finished while the second and third authors were visiting the University of Waterloo. Thanks to Marialuisa de Resmini, Kevin Phelps, and Alex Rosa for valuable insights on the problem. Research of the first and third authors is supported by NSERC Canada under grants A0579 and A9287.
PY - 1991/5
Y1 - 1991/5
N2 - A spanning set in a Steiner triple system is a set of elements for which each element not in the spanning set appears in at least one triple with a pair of elements from the spanning set. A scattering set is a set of elements that is independent, and for which each element not in the scattering set is in at most one triple with a pair of elements from the scattering set. For each v ≡ 1, 3 (mod 6), we exhibit a Steiner triple system with a spanning set of minimum cardinality, and a Steiner triple system with a scattering set of maximum cardinality. In the process, we establish the existence of Steiner triple systems with complete arcs of the minimum possible cardinality.
AB - A spanning set in a Steiner triple system is a set of elements for which each element not in the spanning set appears in at least one triple with a pair of elements from the spanning set. A scattering set is a set of elements that is independent, and for which each element not in the scattering set is in at most one triple with a pair of elements from the scattering set. For each v ≡ 1, 3 (mod 6), we exhibit a Steiner triple system with a spanning set of minimum cardinality, and a Steiner triple system with a scattering set of maximum cardinality. In the process, we establish the existence of Steiner triple systems with complete arcs of the minimum possible cardinality.
UR - http://www.scopus.com/inward/record.url?scp=0013543351&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=0013543351&partnerID=8YFLogxK
U2 - 10.1016/0097-3165(91)90004-Z
DO - 10.1016/0097-3165(91)90004-Z
M3 - Article
AN - SCOPUS:0013543351
SN - 0097-3165
VL - 57
SP - 46
EP - 59
JO - Journal of Combinatorial Theory, Series A
JF - Journal of Combinatorial Theory, Series A
IS - 1
ER -