Cycles in the block-intersection graph of pairwise balanced designs

Cycles in the block-intersection graph of pairwise balanced designs

The focws of this thesis is to investigate certain graph theoretic propertir-s of a class of graphs \shicL arise from combinatorial design theory. The Itlock-i~~tc.rscc.tion graph of a pairwise balanced design has as its vertices tlw Mocks of tllc (hasign ;irici has as its edges precisely those pairs of blocks wliicll llrtvr. nori-cm~pty ilitc~rsc~c'ticm. These graphs have a particular local st...

Some Pairwise Balanced Designs

A pairwise balanced design, B(K; v), is a block design on v points, with block sizes taken from K, and with every pair of points occurring in a unique block; for a fixed K, B(K) is the set of all v for which a B(K; v) exists. A set, S, is a PBD-basis for the set, T , if T = B(S). Let Na(m) = {n : n ≡ a mod m}, and N≥m = {n : n ≥ m}; with Q the corresponding restriction of N to prime powers. Thi...

Pairwise balanced designs with block sizes 4 and 8

Let K be a set of positive integers. A pairwise balanced design (PBD) of index unity B(K,l;v) is a pair (X/~) where X is a v-set (of points) and B is a collection of subsets of X (called blocks) with sizes from K such that every pair of distinct points of X is contained in exactly one block of~. A necessary condition for the existence of a PBD B({4,8},I;v) is v-0 or l(mod 4). It is shown that t...

Pairwise balanced designs of dimension three

The dimension of a pairwise balanced design PBD(v,K) is the maximum positive integer d such that any d of its points are contained in a proper flat. The standard examples are affine and projective linear spaces. Also, Teirlinck provided a nearly complete existence theory for ‘Steiner spaces’, which is the case K = {3} and d = 3. A recent result of the first author and A.C.H. Ling says that ther...

Block Intersection Numbers of Block Designs Ii