On combinatorial designs with subdesigns

Rolf Rees, D.R. Stinson
1989 Discrete Mathematics  
We develop some powerful techniques by which (certain classes of) combinatorial designs with pre-specified subdesigns can be constructed. We use our method to give nearly complete solutions (i.e. to within a finite number of cases) to several problems, including the existence of Kirkman Triple Systems with Subsystems, the existence of (u, 4, l)-BIBDs with subdesigns and the existence of (certain) complementary decompositions with sub-decompositions. 0012-365X/89/33.50 @ 1989, Elsevier Science
more » ... blishers B.V. (North-Holland) decomposition AK, + % is a decomposition 9 of the complete multigraph AK, into K,'s (i.e. a (v, n, A)-BIBD) with the property that for each j = 1, . . . , A the set { Gj E K" : K" E 9} is a decomposition of KU (we will refer to 9 as the root); note that this necessarily means that each Gj E % contains the same number (namely (n(n -1))/2A) of edges. Note that the case A = 1 corresponds to constructing (v, n, l)-BIBDs. Where A > 1 the best-known examples of these designs are the so-called Nested Steiner Triple Systems. A Steiner Triple System STS(v) is said to be nested if one can add a point to each triple in the system and so obtain a (v, 4, 2)-BIBD. The spectrum of these designs was determined by Stinson [32]:
doi:10.1016/0012-365x(89)90365-8 fatcat:fnyn6cyzhjff5fpajz3eqvu2ji