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Tactical configurations: An introduction

  • Judith Q. Longyear
Part III: Contributed Papers New Results On Graphs And Combinatorics
Part of the Lecture Notes in Mathematics book series (LNM, volume 406)

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,...,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.

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References

  1. 1.
    Longyear, J. Q., "Large Tactical Configurations", Discrete Math. 4 (1973) 379–382.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Longyear, J. Q., "Non-Existence Criteria for Small Configurations", Canad. J. Math. 25 (1973) 213–215.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Moore, B. H., "Tactical Memoranda I, II, III", Amer. J. Math. 18 (1896) 254–303.Google Scholar
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    Payne, S. E., and Tinsley, M. F., “On v1 × v2(n,s,t) Configurations”, J. Combinatorial Theory 7 (1969), 1–14.MathSciNetzbMATHCrossRefGoogle Scholar

Applications

  1. 5.
    Busacker, R. G. and Saaty, T. L., Finite Graphs and Networks, McGraw Hill, New York, 1965)zbMATHGoogle Scholar
  2. 6.
    Mycielski, J. and Ulam, S. M., "On the pairing process and the Notion of Generalized Distance", J. Combinatorial Theory 6, p. 227–234.Google Scholar
  3. 7.
    Pless, Vera, "On the Uniqueness of the Golay Codes",J. Combinatorial Theory 5, p. 215–228.Google Scholar

Copyright information

© Springer-Verlag Berlin 1974

Authors and Affiliations

  • Judith Q. Longyear
    • 1
  1. 1.Dartmouth CollegeUSA

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