Togs (generalizations of Two-graphs)

We generalize two-graphs (\unitogs") to \togs", structures consisting of sets of polygons or cycles in a graph that satisfy speciied relations. Togs are equivalent to switching classes of signings of a particular base graph. We develop a mechanism for generating and proving examples and apply it to unitogs (based on complete graphs), bipartite, tripartite, and multi-partite togs (based on complete multipartite graphs), circular togs (based on complete circular multipartite graphs), Hamming togs (based on Hamming graphs), and Johnson togs (based on Johnson graphs, whose nodes are the r-subsets of a set, with adjacency corresponding to (r ? 1)-element overlap). We also explore the possibility of generalizing unitogs to set families which are determined by the sets on any one point. In some examples such a \residually determined" family must be a tog or one of a few exceptional families.