Tactical configurations: An introduction

The original definition of tactical configuration was given by E. H. Moore in 1896, but the definition now in use is in terms of graph theory. A tactical configuration of rank r is a collection of r disjoint vertex sets A1,...,Ar called bands and a relation of incidence among these vertices, so that each vertex in band Ai is incident with the same number of vertices in Aj. This constant number, say di,j, is called the i–j degree, and the collection of all the i–j degrees is called the set of degrees for the configuration. Note that di,j need not be equal to dj,i. The numbers di,i are not defined, since each band is composed of independent vertices. Thus a tactical configuration may be regarded as a multiregular, r-partite graph. The girth of a graph, or of a tactical configuration regarded as a graph, is the number of vertices in any smallest polygon in the graph. This paper describes the important questions concerning the construction and existence of tactical configurations.