Graphical t-wise balanced designs

Abstract

A pair ( X, B ) will be a t-wise balanced design ( tBD) of type t−( v, K, λ) if B = (B i: i ϵ I) is a family of subsets of X, called blocks, such that: (i) |X| = v ϵ N , where N is the set of positive integers; (ii) 1⩽t⩽|B i|ϵK⊆ N , for every i ϵ I; and (iii) if T ⊆ X, | T| = t, then there are λ ϵ N indices i ϵ I where T ⊆ B i . Throughout this paper we make three restrictions on our tBD's: (1) there are no repeated blocks, i.e. B will be a set of subsets of X; (2) t ∉ K or there are no blocks of size t; and (3) P k(X)⊈ B or B does not contain all k-subsets of X for any t< k⩽ v. Note then that X ∉ B . Also, if we give the parameters of a specific tBD, then we will choose a minimal K. We focus on the t−(( p 2 ), K, λ) designs with the symmetric group S p as automorphism group, i.e. X will be the set of v = ( p 2 ) labelled edges of the undirected complete graph K p and if B ϵ B then all subgraphs of K p isomorphic to B are also in B . Call such tBD's ‘graphical tBD's’. We determine all graphical tBD's with λ = 1 or 2 which will include one with parameters 4−(15,{5,7},1).