Ordinary graphs and subplane partitions

We introduce a generalization of symmetric (v, k, λ) block designs, and show how these could potentially be used to construct projective planes of non-prime-power order. If q is a prime power and n2+n+1 = N(q2+q+1), then conceivably we could construct a projective plane of order n by gluing together N planes of order q. For example, 182+18+1 = 343 = 49 ·7. Can we make a projective plane of order 18 by gluing 49 planes of order 2? With this in mind, we will discuss a class of directed graphs which arises when we attempt such a construction. The existence of a graph with the necessary parameters is the first problem we face. Then we will describe how the graphs may be used to build the putative plane of order n. The gluing maps are an even more serious obstacle. Nonetheless, it is an intriguing program, which just possibly could work. 1. Ordinary Graphs and their Associated Matrices Let G be a loopless directed graph. For a vertex i of G, let