Tactical configurations: An introduction

Abstract

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,...,A_r called bands and a relation of incidence among these vertices, so that each vertex in band A_i is incident with the same number of vertices in A_j. This constant number, say d_i,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 d_i,j need not be equal to d_j,i. The numbers d_i,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.