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Combinatorial Configurations

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Abstract

Chapter 5 is devoted to the study of the combinatorial configurations, which is an incidence structure whose incidence graph is bipartite, regular on each bipartition, and having girth six. All the classical geometric point-line configurations are examples of combinatorial configurations, and this chapter is devoted to studying them and their automorphisms and self-dualities using only this combinatorial description.

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References

  1. K. Appel, W. Haken, Every planar map is four colorable. Bull. Am. Math. Soc. 82(5), 711–712 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  2. P.K. Aravind, The generalized kochen-specker theorem. Phys. Rev. A 68, 052104 (2003)

    Article  MathSciNet  Google Scholar 

  3. D.S. Archdeacon, R.B. Richter, The construction and classification of self-dual polyhedra. JCTB 54(1), 37–48 (1992)

    Article  MathSciNet  Google Scholar 

  4. J. Ashley, B. Grünbaum, G.C. Shephard, W. Stromquist, Self-duality groups and ranks of self-dualities. in Applied Geometry and Discrete Mathematics, vol. 4 of DIMACS Series in Discrete Mathamatics and Theoretical Computer Science (American Mathematical Society, Providence, 1991), pp. 11–50

    Google Scholar 

  5. L. Babai, C.D. Godsil, On the automorphism groups of almost all Cayley graphs. Eur. J. Combin. 3(1), 9–15 (1982)

    MathSciNet  MATH  Google Scholar 

  6. D.W. Barnette, B. Grünbaum, On Steinitz’s theorem concerning convex 3-polytopes and on some properties of planar graphs. in The Many Facets of Graph Theory (Proc. Conf., Western Mich. Univ., Kalamazoo, Mich., 1968) (Springer, Berlin, 1969), pp. 27–40

    Google Scholar 

  7. A. Betten, G. Brinkmann, T. Pisanski, Counting symmetric configurations v 3. in Proceedings of the 5th Twente Workshop on Graphs and Combinatorial Optimization (Enschede, 1997), Discrete Appl. Math. 99(1–3), 331–338, (2000)

    Google Scholar 

  8. N. Biggs, Algebraic Graph Theory, 2nd edn. (Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1993)

    Google Scholar 

  9. M. Boben, T. Pisanski, A. Žitnik, I-graphs and the corresponding configurations. J. Combin. Des. 13(6), 406–424 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Bokowski, B. Sturmfels, Computational Synthetic Geometry, vol. 1355 of Lecture Notes in Mathematics (Springer, Berlin, 1989)

    Google Scholar 

  11. I.Z. Bouwer, An edge but not vertex transitive cubic graph. Bull. Can. Math. Soc. 11, 533–535 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  12. I.Z. Bouwer, On edge but not vertex transitive regular graphs. J. Combin. Theor. B 12(1), 32–40 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  13. G. Brinkmann, B.D. McKay, C. Saager, The smallest cubic graphs of girth nine. Combin. Probab. Comput. 4(4), 317–329 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. R.H. Bruck, H.J. Ryser, The nonexistence of certain finite projective planes. Can. J. Math. 1, 88–93 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  15. S. Chowla, H.J. Ryser, Combinatorial problems. Can. J. Math. 2, 93–99 (1950)

    MathSciNet  MATH  Google Scholar 

  16. H.S.M. Coxeter, Self-dual configurations and regular graphs. Bull. Am. Math. Soc. 56, 413–455 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  17. H.S.M. Coxeter, Twelve Geometric Essays (Southern Illinois University Press, Carbondale, 1968)

    MATH  Google Scholar 

  18. H.S.M. Coxeter, Projective Geometry (Springer, New York, 1994); Revised reprint of the second (1974) edition

    Google Scholar 

  19. H.S.M. Coxeter, in Coloured Symmetry, ed. by H.S.M. Coxeter et al. M. C. Escher: Art and Science (Elsevier, Amsterdam, 1986), pp. 15–33

    Google Scholar 

  20. H.S.M. Coxeter, R. Frucht, D.L. Powers, Zero-Symmetric Graphs (Academic [Harcourt Brace Jovanovich Publishers], New York, 1981), Trivalent graphical regular representations of groups

    Google Scholar 

  21. H.S.M. Coxeter, W.O.J. Moser, in Generators and Relations for Discrete Groups, vol. 14 of Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 4th edn. (Springer, Berlin, 1980)

    Google Scholar 

  22. P.R. Cromwell, Polyhedra (Cambridge University Press, Cambridge, 1997)

    MATH  Google Scholar 

  23. W.H. Cunningham, J. Edmonds, A combinatorial decomposition theory. Can. J. Math. 32(3), 734–765 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  24. R.D. von Sterneck, Die configurationen 113. Monatsh. Math. Phys. 5(1), 325–330 (1894)

    Article  MathSciNet  MATH  Google Scholar 

  25. R. von Sterneck, Die configurationen 123. Monatsh. Math. Phys. 6(1), 223–254 (1895)

    Article  MathSciNet  MATH  Google Scholar 

  26. A. Deza, M. Deza, V. Grishukhin, Fullerenes and coordination polyhedra versus half-cube embeddings. Discrete Math. 192(1–3), 41–80 (1998). Discrete metric spaces (Villeurbanne, 1996)

    Google Scholar 

  27. C. Droms, B. Servatius, H. Servatius, The structure of locally finite two-connected graphs. Electr. J. Comb. 2 (1995)

    Google Scholar 

  28. C. Droms, B. Servatius, H. Servatius, Connectivity and planarity of Cayley graphs. Beiträge Algebra Geom. 39(2), 269–282 (1998)

    MathSciNet  MATH  Google Scholar 

  29. D. Eppstein, Finding large clique minors is hard. J. Graph Algorithms Appl. 13(2), 197–204 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. L. Euler, Recherches sur une nouvelle espece de quarres magiques, in Verhandelingen Uitgegeven Door het Zeeuwsch Genootschap der Wetenschappen te Vlissingen 9, Middelburg, 1782, pp. 85–239

    Google Scholar 

  31. J. Folkman, Regular line-symmetric graphs. J. Combin. Theor. 3, 215–232 (1967)

    MathSciNet  MATH  Google Scholar 

  32. R.M. Foster, The Foster Census (Charles Babbage Research Centre, Winnipeg, 1988); R. M. Foster’s census of connected symmetric trivalent graphs, With a foreword by H. S. M. Coxeter, With a biographical preface by Seymour Schuster, With an introduction by I. Z. Bouwer, W. W. Chernoff, B. Monson and Z. Star, Edited and with a note by Bouwer

    Google Scholar 

  33. R. Frucht, A canonical representation of trivalent Hamiltonian graphs. J. Graph Theor. 1(1), 45–60 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  34. R. Frucht, J.E. Graver, M.E. Watkins, The groups of the generalized Petersen graphs. Proc. Camb. Philos. Soc. 70, 211–218 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  35. C. Godsil, G. Royle, in Algebraic Graph Theory, vol. 207 of Graduate Texts in Mathematics (Springer, New York, 2001)

    Google Scholar 

  36. M. Goldberg, A class of multi-symmetric polyhedra. Tohoku Math. J. 43, 104–108 (1937)

    Google Scholar 

  37. H. Gropp, in On the History of Configurations, ed. by A. Deza, J. Echeverria, A. Ibarra. International Symposium on Structures in Mathematical Theories (University del Pais Vasco, Bilbao, 1990), pp. 263–268

    Google Scholar 

  38. H. Gropp, On the existence and nonexistence of configurations n k . J. Combin. Inform. Syst. Sci. 15(1–4), 34–48 (1990); Graphs, designs and combinatorial geometries (Catania, 1989)

    Google Scholar 

  39. H. Gropp, Configurations and graphs. Discrete Math. 111(1–3), 269–276 (1993); Graph theory and combinatorics (Marseille-Luminy, 1990)

    Google Scholar 

  40. H. Gropp, The construction of all configurations (124, 163), in Fourth Czechoslovakian Symposium on Combinatorics, Graphs and Complexity (Prachatice, 1990), vol. 51 of Annals in Discrete Mathetics (North-Holland, Amsterdam, 1992), pp. 85–91

    Google Scholar 

  41. H. Gropp, Configurations and their realization. Discrete Math. 174(1–3), 137–151 (1997); Combinatorics (Rome and Montesilvano, 1994)

    Google Scholar 

  42. J.L. Gross, T.W. Tucker, Topological Graph Theory (Dover, Mineola, 2001); Reprint of the 1987 original [Wiley, New York; MR0898434 (88h:05034)] with a new preface and supplementary bibliography

    Google Scholar 

  43. B. Grünbaum, (1–2–3)-complexes. Geombinatorics 13(2), 65–72 (2003)

    Google Scholar 

  44. B. Grünbaum, in Configurations of Points and Lines, vol. 103 of Graduate Studies in Mathematics (American Mathematical Society, Providence, 2009)

    Google Scholar 

  45. B. Grünbaum, J.F. Rigby, The real configuration (214). J. Lond. Math. Soc. (2) 41(2), 336–346 (1990)

    Google Scholar 

  46. B. Grunbaum, G.C. Shephard, Is selfduality involutory? Am. Math. Mon 95(8), 729–733 (1988)

    Article  Google Scholar 

  47. B. Grünbaum, G.C. Shephard, Tilings and Patterns. A Series of Books in the Mathematical Sciences (W. H. Freeman and Company, New York, 1989); An introduction

    Google Scholar 

  48. B. Grünbaum, G.C. Shephard, Isohedra with dart-shaped faces. Discrete Math. 241(1–3), 313–332 (2001); Selected papers in honor of Helge Tverberg

    Google Scholar 

  49. W.H. Haemers, D.G. Higman, S.A. Hobart, in Strongly Regular Graphs Induced by Polarities of Symmetric Designs. Advances in Finite Geometries and Designs (Chelwood Gate, 1990) (Oxford Sci. Publ., Oxford University Press, New York, 1991), pp. 163–168

    Google Scholar 

  50. D. Hilbert, S.Cohn-Vossen, Geometry and the Imagination (Chelsea Publishing Company, New York, 1952); Translated by P. Neményi

    Google Scholar 

  51. D. Hilbert, S. Cohn-Vossen, Anschauliche Geometrie (Wissenschaftliche Buchgesellschaft, Darmstadt, 1973); Mit einem Anhang: “Einfachste Grundbegriffe der Topologie” von Paul Alexandroff, Reprint der 1932 Ausgabe

    Google Scholar 

  52. M. Hladnik, D. Marušič, T. Pisanski, Cyclic Haar graphs. Discrete Math. 244(1–3), 137–152 (2002); Algebraic and topological methods in graph theory (Lake Bled, 1999)

    Google Scholar 

  53. J.E. Hopcroft, R.E. Tarjan, Dividing a graph into triconnected components. SIAM J. Comput. 2, 135–158 (1973)

    Article  MathSciNet  Google Scholar 

  54. S. Jendroľ, On symmetry groups of self-dual convex polyhedra, in Fourth Czechoslovakian Symposium on Combinatorics, Graphs and Complexity (Prachatice, 1990), vol. 51 of Annals of Discrete Mathematics (North-Holland, Amsterdam, 1992), pp. 129–135

    Google Scholar 

  55. T.P. Kirkman, On autopolar polyhedra. Philos. Trans. Roy. Soc. Lond. 147, 183–215 (1857)

    Article  Google Scholar 

  56. H.S. Koike, I. Kovács, T. Pisanski, Enumeration of cyclic n 3 and n 4 configurations. Isomorphic tetravalent cyclic Haar graphs to appear in Ars Mathematica Contemporanea, (2014)

    Google Scholar 

  57. I. Kovács, M. Servatius, On cayley digraphs on nonisomorphic 2-groups. J. Graph Theor. 70(4), 435–448 (2012)

    Article  MATH  Google Scholar 

  58. C.W.H. Lam, G. Kolesova, L. Thiel, A computer search for finite projective planes of order 9. Discrete Math. 92(1–3), 187–195 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  59. C.W.H. Lam, L. Thiel, S. Swiercz, The nonexistence of finite projective planes of order 10. Can. J. Math. 41(6), 1117–1123 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  60. F.W. Levi, Geometrische Konfigurationen. Mit einer Einführung in die Kombinatorische Flächentopologie (S. Hirzel, Leipzig, 1929)

    Google Scholar 

  61. W. Magnus, A. Karrass, D. Solitar, Combinatorial Group Theory, 2nd edn. (Dover, Mineola, 2004); Presentations of groups in terms of generators and relations

    Google Scholar 

  62. A. Malnič, D. Marušič, P. Potočnik, C. Wang, An infinite family of cubic edge- but not vertex-transitive graphs. Discrete Math. 280(1–3), 133–148 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  63. D. Marušič, T. Pisanski, Weakly flag-transitive configurations and half-arc-transitive graphs. Eur. J. Combin. 20(6), 559–570 (1999)

    Article  MATH  Google Scholar 

  64. D. Marušič, T. Pisanski, The remarkable generalized petersen graph gp(8, 3). Math. Slovaca 50, 117–121 (2000)

    MathSciNet  MATH  Google Scholar 

  65. D. Marušič, T. Pisanski, The Gray graph revisited. J. Graph Theor. 35(1), 1–7 (2000)

    Article  MATH  Google Scholar 

  66. D. Marušič, T. Pisanski, S. Wilson, The genus of the gray graph is 7. Eur. J. Combin. 26(3–4), 377–385 (2005)

    MATH  Google Scholar 

  67. D. Marušič, M.-Y. Xu, A \(\frac{1}{2}\)-transitive graph of valency 4 with a nonsolvable group of automorphisms. J. Graph Theor. 25(2), 133–138 (1997)

    Article  MATH  Google Scholar 

  68. D. Marušič, On vertex symmetric digraphs. Discrete Math. 36, 69–81 (1981)

    MATH  Google Scholar 

  69. B. Mohar, P. Rosenstiehl, Tessellation and visibility representations of maps on the torus. Discrete Comput. Geom. 19(2), 249–263 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  70. B. Monson, T. Pisanski, E. Schulte, A.I. Weiss, Semisymmetric graphs from polytopes. J. Combin. Theor. Ser. A 114(3), 421–435 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  71. F.R. Moulton, A simple non-Desarguesian plane geometry. Trans. Am. Math. Soc. 3(2), 192–195 (1902)

    Article  MathSciNet  MATH  Google Scholar 

  72. R. Nedela, M. Škoviera, Which generalized Petersen graphs are Cayley graphs? J. Graph Theor. 19(1), 1–11 (1995)

    Article  MATH  Google Scholar 

  73. P.M. Neumann, A lemma that is not Burnside’s. Math. Sci. 4(2), 133–141 (1979)

    MathSciNet  Google Scholar 

  74. A. Orbanić, M. Petkovšek, T. Pisanski, P. Potočnik, A note on enumeration of one-vertex maps. Ars Math. Contemp. 3(1), 1–12 (2010)

    MathSciNet  MATH  Google Scholar 

  75. M. Petkovšek, H. Zakrajšek, Enumeration of I-graphs: Burnside does it again. Ars Math. Contemp. 2(2), 241–262 (2009)

    MathSciNet  MATH  Google Scholar 

  76. T. Pisanski, M. Boben, D. Marušič, A. Orbanić, A. Graovac, The 10-cages and derived configurations. Discrete Math. 275(1–3), 265–276 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  77. T. Pisanski, A. Žitnik, A. Graovac, A. Baumgartner, Rotagraphs and their generalizations. J. Chem. Inform. Comput. Sci. 34(5), 1090–1093 (1994)

    Article  Google Scholar 

  78. T. Pisanski, A classification of cubic bicirculants. Discrete Math. 307(3–5), 567–578 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  79. T. Pisanski, Yet another look at the Gray graph. New Zealand J. Math. 36, 85–92 (2007)

    MathSciNet  MATH  Google Scholar 

  80. T. Pisanski, M. Randić, Bridges between geometry and graph theory, in Geometry at Work, vol. 53 of MAA Notes (Mathematical Assocition of America, Washington, DC, 2000), pp. 174–194

    Google Scholar 

  81. T. Pisanski, D. Schattschneider, B. Servatius, Applying Burnside’s lemma to a one-dimensional Escher problem. Math. Mag. 79(3), 167–180 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  82. G. Ringel, Map Color Theorem (Springer, New York, 1974); Die Grundlehren der mathematischen Wissenschaften, Band 209

    Google Scholar 

  83. J.J. Rotman, in An Introduction to the Theory of Groups, vol. 148 of Graduate Texts in Mathematics, 4th edn. (Springer, New York, 1995)

    Google Scholar 

  84. C.P. Rourke, B.J. Sanderson, Introduction to Piecewise-Linear Topology (Springer, New York, 1972)

    Book  MATH  Google Scholar 

  85. G. Sabidussi, Graphs with given group and given graph-theoretical properties. Can. J. Math. 9, 515–525 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  86. G. Salmon, A Treatise on Conic Sections. 6’th ed. New York Chelsea Pub., (1954) Reprinted by the American Mathematical Society (Providence, Rhode Island, 2005)

    Google Scholar 

  87. D. Schattschneider, Escher’s combinatorial patterns. Electron. J. Combin. 4(2) (1997); Research Paper 17, approx. 31 pp. (electronic). The Wilf Festschrift (Philadelphia, PA, 1996)

    Google Scholar 

  88. A.E. Schroth, How to draw a hexagon. Discrete Math. 199(1–3), 161–171 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  89. B. Servatius, H. Servatius, The 24 symmetry pairings of self–dual maps on the sphere. Discrete Math. 140, 167–183 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  90. B. Servatius, H. Servatius, Self-dual graphs. Discrete Math. 149(1–3), 223–232 (1996)

    MathSciNet  MATH  Google Scholar 

  91. B. Servatius, H. Servatius, in Symmetry, Automorphisms, and Self-duality of Infinite Planar Graphs and Tilings, ed. by V. Balint. Proceedings of the Internatinal Geometry Conference in Zilina, 1998, pp. 83–116

    Google Scholar 

  92. R.P. Stanley, in Enumerative Combinatorics. Vol. 2, vol. 62 of Cambridge Studies in Advanced Mathematics (Cambridge University Press, Cambridge, 1999); With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin

    Google Scholar 

  93. A. Steimle, W. Staton, The isomorphism classes of the generalized Petersen graphs. Discrete Math. 309(1), 231–237 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  94. E. Steinitz, Über die construction der configurationen n 3. Ph.D. thesis, Kgl. Universität Breslau, 1894

    Google Scholar 

  95. E. Steinitz, H. Rademacher, Vorlesungen über die Theorie der Polyeder unter Einschluss der Elemente der Topologie (Springer, Berlin, 1976); Reprint der 1934 Auflage, Grundlehren der Mathematischen Wissenschaften, No. 41

    Google Scholar 

  96. I. Stewart, Galois Theory, 3rd edn. (Chapman & Hall/CRC Mathematics. Chapman & Hall/CRC, Boca Raton, 2004)

    MATH  Google Scholar 

  97. B. Sturmfels, N. White, All 113 and 123-configurations are rational. Aequationes Math. 39(2–3), 254–260 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  98. W.T. Tutte, A family of cubical graphs. Proc. Camb. Philos. Soc. 43, 459–474 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  99. W.T. Tutte, A census of planar maps. Can. J. Math. 15, 249–271 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  100. W.T. Tutte, How to draw a graph. Proc. Lond. Math. Soc. (3) 13, 743–767 (1963)

    Google Scholar 

  101. W.T. Tutte, Connectivity in Graphs. Mathematical Expositions, No. 15 (University of Toronto Press, Toronto, 1966)

    Google Scholar 

  102. V.G. Vizing, On an estimate of the chromatic class of a p-graph. Diskret. Analiz No. 3, 25–30 (1964)

    MathSciNet  Google Scholar 

  103. D. Wells, The Penguin Dictionary of Curious and Interesting Geometry (Penguin Books, New York, 1991)

    MATH  Google Scholar 

  104. A.T. White, in Graphs, Groups and Surfaces, vol. 8 of North-Holland Mathematics Studies, 2nd edn. (North-Holland Publishing Co., Amsterdam, 1984)

    Google Scholar 

  105. W. Whiteley, A matroid on hypergraphs, with applications in scene analysis and geometry. Discrete Comput. Geom. 4(1), 75–95 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  106. H. Whitney, Congruent graphs and the connectivity of graphs. Am. J. Math. 54(1), 150–168 (1932)

    Article  MathSciNet  Google Scholar 

  107. H. Whitney, 2-Isomorphic Graphs. Am. J. Math. 55(1–4), 245–254 (1933)

    Article  MathSciNet  Google Scholar 

  108. P.K. Wong, Cages—a survey. J. Graph Theor. 6(1), 1–22 (1982)

    Article  MATH  Google Scholar 

  109. E.M. Wright, Burnside’s lemma: A historical note. J. Combin. Theor. Ser. B 30(1), 89–90 (1981)

    Article  MATH  Google Scholar 

  110. K. Zindler, Zur Theorie der Netze und Configurationen. [J] Wien. Ber., Math. Naturw. Kl. 98, 499–519 (1888)

    Google Scholar 

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Pisanski, T., Servatius, B. (2013). Combinatorial Configurations. In: Configurations from a Graphical Viewpoint. Birkhäuser Advanced Texts Basler Lehrbücher. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8364-1_5

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