Graphs and Combinatorics pp 325-329 | Cite as

# 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 A_{1},...,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.

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## Applications

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