Menu
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. \( \lambda{K}_n^* \)) 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 \( \lambda \geq 2 \) and for directed hinge systems for all indices \( \lambda \geq 1 \).