Balanced House-systems and Nestings

Abstract

A \(G\)-design is called balanced if the degree of each vertex \(x\) is a constant. A \(G\)-design is called strongly balanced if for every \(i = 1, 2, \ldots, h\), there exists a constant \(C_i\) such that \(d_{A_i}(x) = C_i\) for every vertex \(x\), where \(A_i\) are the orbits of the automorphism group of \(G\) on its vertex-set and \(d_{A_i}(x)\) of a vertex is the number of blocks containing \(x\) as an element of \(A_i\). We say that a \(G\)-design is simply balanced if it is balanced, but not strongly balanced. In this paper, we determine the spectrum for simply balanced and strongly balanced House-systems. Further, we determine the spectrum for House-systems of all admissible indices nesting \(C_4\)-systems.