Hinge Systems-Spectra and Embeddings

Abstract

We define the $B_2$ block-intersection graph of a balanced incomplete block design $(V,\mathfrak{B})$ having order $n$, block size $k$, and index $\lambda$, or BIBD$(n,k,\lambda )$, to be the graph with vertex set $\mathfrak{B}$ in which two vertices are adjacent if and only if their corresponding blocks have exactly two points of $V$ in common. We define an undirected (resp. directed) hinge to be the multigraph with four vertices which consists of two undirected (resp. directed) 3-cycles which share exactly two vertices in common. An undirected (resp. directed) hinge system of order $n$ and index $\lambda$ is a decomposition of $\lambda K_n$ (resp. $\lambda K_n^{\ast }$) into undirected (resp. directed) hinges. In this paper, we show that each component of the $B_2$ block-intersection graph of a simple BIBD$(n,3,2)$ is 2-edge-connected; this enables us to decompose pure Mendelsohn triple systems and simple 2-fold triple systems into directed and undirected hinge systems, respectively. Furthermore, we obtain a generalisation of the Doyen-Wilson theorem by giving necessary and sufficient conditions for embedding undirected (resp. directed) hinge systems of order $n$ in undirected (resp. directed) hinge systems of order $v$. Finally, we determine the spectrum for undirected hinge systems for all indices $\lambda \ge 2$ and for directed hinge systems for all indices $\lambda \ge 1$.