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 $i=1,2,\dots ,h$, 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.