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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K. Appel, W. Haken, Every planar map is four colorable. Bull. Am. Math. Soc. 82(5), 711–712 (1976)
P.K. Aravind, The generalized kochen-specker theorem. Phys. Rev. A 68, 052104 (2003)
D.S. Archdeacon, R.B. Richter, The construction and classification of self-dual polyhedra. JCTB 54(1), 37–48 (1992)
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
L. Babai, C.D. Godsil, On the automorphism groups of almost all Cayley graphs. Eur. J. Combin. 3(1), 9–15 (1982)
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
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)
N. Biggs, Algebraic Graph Theory, 2nd edn. (Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1993)
M. Boben, T. Pisanski, A. Žitnik, I-graphs and the corresponding configurations. J. Combin. Des. 13(6), 406–424 (2005)
J. Bokowski, B. Sturmfels, Computational Synthetic Geometry, vol. 1355 of Lecture Notes in Mathematics (Springer, Berlin, 1989)
I.Z. Bouwer, An edge but not vertex transitive cubic graph. Bull. Can. Math. Soc. 11, 533–535 (1968)
I.Z. Bouwer, On edge but not vertex transitive regular graphs. J. Combin. Theor. B 12(1), 32–40 (1972)
G. Brinkmann, B.D. McKay, C. Saager, The smallest cubic graphs of girth nine. Combin. Probab. Comput. 4(4), 317–329 (1995)
R.H. Bruck, H.J. Ryser, The nonexistence of certain finite projective planes. Can. J. Math. 1, 88–93 (1949)
S. Chowla, H.J. Ryser, Combinatorial problems. Can. J. Math. 2, 93–99 (1950)
H.S.M. Coxeter, Self-dual configurations and regular graphs. Bull. Am. Math. Soc. 56, 413–455 (1950)
H.S.M. Coxeter, Twelve Geometric Essays (Southern Illinois University Press, Carbondale, 1968)
H.S.M. Coxeter, Projective Geometry (Springer, New York, 1994); Revised reprint of the second (1974) edition
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
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
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)
P.R. Cromwell, Polyhedra (Cambridge University Press, Cambridge, 1997)
W.H. Cunningham, J. Edmonds, A combinatorial decomposition theory. Can. J. Math. 32(3), 734–765 (1980)
R.D. von Sterneck, Die configurationen 113. Monatsh. Math. Phys. 5(1), 325–330 (1894)
R. von Sterneck, Die configurationen 123. Monatsh. Math. Phys. 6(1), 223–254 (1895)
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)
C. Droms, B. Servatius, H. Servatius, The structure of locally finite two-connected graphs. Electr. J. Comb. 2 (1995)
C. Droms, B. Servatius, H. Servatius, Connectivity and planarity of Cayley graphs. Beiträge Algebra Geom. 39(2), 269–282 (1998)
D. Eppstein, Finding large clique minors is hard. J. Graph Algorithms Appl. 13(2), 197–204 (2009)
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
J. Folkman, Regular line-symmetric graphs. J. Combin. Theor. 3, 215–232 (1967)
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
R. Frucht, A canonical representation of trivalent Hamiltonian graphs. J. Graph Theor. 1(1), 45–60 (1977)
R. Frucht, J.E. Graver, M.E. Watkins, The groups of the generalized Petersen graphs. Proc. Camb. Philos. Soc. 70, 211–218 (1971)
C. Godsil, G. Royle, in Algebraic Graph Theory, vol. 207 of Graduate Texts in Mathematics (Springer, New York, 2001)
M. Goldberg, A class of multi-symmetric polyhedra. Tohoku Math. J. 43, 104–108 (1937)
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
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)
H. Gropp, Configurations and graphs. Discrete Math. 111(1–3), 269–276 (1993); Graph theory and combinatorics (Marseille-Luminy, 1990)
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
H. Gropp, Configurations and their realization. Discrete Math. 174(1–3), 137–151 (1997); Combinatorics (Rome and Montesilvano, 1994)
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
B. Grünbaum, (1–2–3)-complexes. Geombinatorics 13(2), 65–72 (2003)
B. Grünbaum, in Configurations of Points and Lines, vol. 103 of Graduate Studies in Mathematics (American Mathematical Society, Providence, 2009)
B. Grünbaum, J.F. Rigby, The real configuration (214). J. Lond. Math. Soc. (2) 41(2), 336–346 (1990)
B. Grunbaum, G.C. Shephard, Is selfduality involutory? Am. Math. Mon 95(8), 729–733 (1988)
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
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
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
D. Hilbert, S.Cohn-Vossen, Geometry and the Imagination (Chelsea Publishing Company, New York, 1952); Translated by P. Neményi
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
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)
J.E. Hopcroft, R.E. Tarjan, Dividing a graph into triconnected components. SIAM J. Comput. 2, 135–158 (1973)
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
T.P. Kirkman, On autopolar polyhedra. Philos. Trans. Roy. Soc. Lond. 147, 183–215 (1857)
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)
I. Kovács, M. Servatius, On cayley digraphs on nonisomorphic 2-groups. J. Graph Theor. 70(4), 435–448 (2012)
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)
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)
F.W. Levi, Geometrische Konfigurationen. Mit einer Einführung in die Kombinatorische Flächentopologie (S. Hirzel, Leipzig, 1929)
W. Magnus, A. Karrass, D. Solitar, Combinatorial Group Theory, 2nd edn. (Dover, Mineola, 2004); Presentations of groups in terms of generators and relations
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)
D. Marušič, T. Pisanski, Weakly flag-transitive configurations and half-arc-transitive graphs. Eur. J. Combin. 20(6), 559–570 (1999)
D. Marušič, T. Pisanski, The remarkable generalized petersen graph gp(8, 3). Math. Slovaca 50, 117–121 (2000)
D. Marušič, T. Pisanski, The Gray graph revisited. J. Graph Theor. 35(1), 1–7 (2000)
D. Marušič, T. Pisanski, S. Wilson, The genus of the gray graph is 7. Eur. J. Combin. 26(3–4), 377–385 (2005)
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)
D. Marušič, On vertex symmetric digraphs. Discrete Math. 36, 69–81 (1981)
B. Mohar, P. Rosenstiehl, Tessellation and visibility representations of maps on the torus. Discrete Comput. Geom. 19(2), 249–263 (1998)
B. Monson, T. Pisanski, E. Schulte, A.I. Weiss, Semisymmetric graphs from polytopes. J. Combin. Theor. Ser. A 114(3), 421–435 (2007)
F.R. Moulton, A simple non-Desarguesian plane geometry. Trans. Am. Math. Soc. 3(2), 192–195 (1902)
R. Nedela, M. Škoviera, Which generalized Petersen graphs are Cayley graphs? J. Graph Theor. 19(1), 1–11 (1995)
P.M. Neumann, A lemma that is not Burnside’s. Math. Sci. 4(2), 133–141 (1979)
A. Orbanić, M. Petkovšek, T. Pisanski, P. Potočnik, A note on enumeration of one-vertex maps. Ars Math. Contemp. 3(1), 1–12 (2010)
M. Petkovšek, H. Zakrajšek, Enumeration of I-graphs: Burnside does it again. Ars Math. Contemp. 2(2), 241–262 (2009)
T. Pisanski, M. Boben, D. Marušič, A. Orbanić, A. Graovac, The 10-cages and derived configurations. Discrete Math. 275(1–3), 265–276 (2004)
T. Pisanski, A. Žitnik, A. Graovac, A. Baumgartner, Rotagraphs and their generalizations. J. Chem. Inform. Comput. Sci. 34(5), 1090–1093 (1994)
T. Pisanski, A classification of cubic bicirculants. Discrete Math. 307(3–5), 567–578 (2007)
T. Pisanski, Yet another look at the Gray graph. New Zealand J. Math. 36, 85–92 (2007)
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
T. Pisanski, D. Schattschneider, B. Servatius, Applying Burnside’s lemma to a one-dimensional Escher problem. Math. Mag. 79(3), 167–180 (2006)
G. Ringel, Map Color Theorem (Springer, New York, 1974); Die Grundlehren der mathematischen Wissenschaften, Band 209
J.J. Rotman, in An Introduction to the Theory of Groups, vol. 148 of Graduate Texts in Mathematics, 4th edn. (Springer, New York, 1995)
C.P. Rourke, B.J. Sanderson, Introduction to Piecewise-Linear Topology (Springer, New York, 1972)
G. Sabidussi, Graphs with given group and given graph-theoretical properties. Can. J. Math. 9, 515–525 (1957)
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)
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)
A.E. Schroth, How to draw a hexagon. Discrete Math. 199(1–3), 161–171 (1999)
B. Servatius, H. Servatius, The 24 symmetry pairings of self–dual maps on the sphere. Discrete Math. 140, 167–183 (1995)
B. Servatius, H. Servatius, Self-dual graphs. Discrete Math. 149(1–3), 223–232 (1996)
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
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
A. Steimle, W. Staton, The isomorphism classes of the generalized Petersen graphs. Discrete Math. 309(1), 231–237 (2009)
E. Steinitz, Über die construction der configurationen n 3. Ph.D. thesis, Kgl. Universität Breslau, 1894
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
I. Stewart, Galois Theory, 3rd edn. (Chapman & Hall/CRC Mathematics. Chapman & Hall/CRC, Boca Raton, 2004)
B. Sturmfels, N. White, All 113 and 123-configurations are rational. Aequationes Math. 39(2–3), 254–260 (1990)
W.T. Tutte, A family of cubical graphs. Proc. Camb. Philos. Soc. 43, 459–474 (1947)
W.T. Tutte, A census of planar maps. Can. J. Math. 15, 249–271 (1963)
W.T. Tutte, How to draw a graph. Proc. Lond. Math. Soc. (3) 13, 743–767 (1963)
W.T. Tutte, Connectivity in Graphs. Mathematical Expositions, No. 15 (University of Toronto Press, Toronto, 1966)
V.G. Vizing, On an estimate of the chromatic class of a p-graph. Diskret. Analiz No. 3, 25–30 (1964)
D. Wells, The Penguin Dictionary of Curious and Interesting Geometry (Penguin Books, New York, 1991)
A.T. White, in Graphs, Groups and Surfaces, vol. 8 of North-Holland Mathematics Studies, 2nd edn. (North-Holland Publishing Co., Amsterdam, 1984)
W. Whiteley, A matroid on hypergraphs, with applications in scene analysis and geometry. Discrete Comput. Geom. 4(1), 75–95 (1989)
H. Whitney, Congruent graphs and the connectivity of graphs. Am. J. Math. 54(1), 150–168 (1932)
H. Whitney, 2-Isomorphic Graphs. Am. J. Math. 55(1–4), 245–254 (1933)
P.K. Wong, Cages—a survey. J. Graph Theor. 6(1), 1–22 (1982)
E.M. Wright, Burnside’s lemma: A historical note. J. Combin. Theor. Ser. B 30(1), 89–90 (1981)
K. Zindler, Zur Theorie der Netze und Configurationen. [J] Wien. Ber., Math. Naturw. Kl. 98, 499–519 (1888)
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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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