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Theory of Finite and Infinite Graphs

  • Dénes König

Abstract

Let {A, B, C…} be a set of “points.” If certain pairs of these points are connected by one or more “lines”, the resulting configuration is called a graph. Those points of {A, B, C…} which are connected with at least one point are called vertices of the graph. (Vertices which could be called “isolated” are therefore excluded.) The lines involved are called edges of the graph1. An edge which connects A and B, i.e. whose endpoints are A and B, and which goes to A (and B) we shall designate by AB. It is possible that several edges are designated as AB. If A is an endpoint of edge k, we shall also say that A and k are incident to each other. If the set of vertices and the set of edges of a graph are both finite, the graph is called finite, otherwise infinite. An infinite graph has infinitely many edges but possibly only finitely many vertices (e.g., two vertices can be connected by infinitely many edges.)

Keywords

Bipartite Graph Star Form Regular Graph Hamiltonian Cycle Finite Graph 
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Bibliography

  1. [1]
    Ahrens (W), Über das Gleichungssystem einer Kirchhoffschen galvanischen Stromver(weigung, Mathematische Annalen, 49, 1897, pp. 311–324.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Ahrens (W), Mathematische Unterhaltungen und Spiele, Leip(ig, 1st edition, 1901; latest edition: Vol. 1, 3rd edition, 1921; Vol. II, 2nd edition, 1918Google Scholar
  3. [1]
    Alexander (J.W), Combinatorial Analysis Situs, Transactions of the American Mathematical Society, 28, 1926, pp. 301–329.MathSciNetMATHCrossRefGoogle Scholar
  4. Alexander (J.W.), (see Vehlen).Google Scholar
  5. [1]
    Baker (R.P), Cayley diagrams on the anchor ring, American Journal of Mathematics, 53, 1931, pp. 645–669.MathSciNetCrossRefGoogle Scholar
  6. [1]
    Ball (Rouse W.) Mathematical Recreations and Problems, 1st edition London 1892. Latest French edition: Récréations mathématiques et problèmes des temps anciens et modernes, traduit par J. Fit - Patrick, Vols. I-III, Paris 1926/27.Google Scholar
  7. [1]
    Baltzer (R,) Eine Erinnerung an Möbius und seinen Freund Weiske, Berichte der k. sächsischen Gesellschaft der Wissenschaften, 37, 1885, pp. 1–6.Google Scholar
  8. [1]
    Birkhoff (G.D,) On the number of ways of colouring a map. Proceedings of the Edinbourgh Mathematical Society, (2,) part II, 1930, pp. 83–91.CrossRefGoogle Scholar
  9. [1]
    Brahana (H.R)., A proof of Petersen’s theorem. Annals of Mathematics (2,) 19, 1917, pp. 59–63.MathSciNetMATHCrossRefGoogle Scholar
  10. Brunel (G.), [1] (Sur un problème de combinaisons,) Extraits des Procés - Verbaux des séances de la Société des Sciences de Bordeaux, 1893/94, pp. XIV-XV.Google Scholar
  11. [2]
    Brunel (G.), Analysis Situs. Recherches sur les réseaux. Mémoires de la Société des Sciences de Bordeaux (4,) 5, 1895, pp. 165–21S.Google Scholar
  12. [3]
    Brunel (G)., Construction d’un réseau donné l’aide d’un nombre déterminé de traits. Procés - Verbaux des séances de la Société des Sciences de Bordeaux, 1895/1896, pp. 62–65.Google Scholar
  13. [1]
    Cayley (A.), On the theory of the analytical forms called trees. Philosophical Magazine, 13, 1857, pp. 172–176 - Mathematical Papers, Cambridge 1889–1897, Vol. Ill, pp. 242–246.Google Scholar
  14. [2]
    Cayley (A)., On the mathematical theory of isomers. Philosophical Magazine, 47, 1874 pp. 444–446 = Mathematical Papers, Vol. IX, pp. 202–204.Google Scholar
  15. [3]
    Cayley (A)., On the analytical forms called trees, with application to the theory of chemical combinations. Report of the British Association for the Advancement of Science, 1875, pp. 257–305 = Mathematical Papers. Vol. IX, pp. 427–460Google Scholar
  16. [4]
    Cayley (A)., Solution of problem 5208, Mathematical Questions with Solutions from the Educational Times, 27, 1877, pp. 81–83 = Mathematical Papers, Vol. X, pp. 598–600.Google Scholar
  17. [5]
    Cayley (A)., Desiderata and suggestions. Nr. 2: The theory of groups, graphical representation, American Journal of Mathematics, 1, 1878, pp. 174–176 = Mathematical Papers, Vol. X, pp. 403–405.MathSciNetGoogle Scholar
  18. [6]
    Cayley (A)., On the theory of groups. Proceedings of the London Mathematical Society, 9, 1878, pp. 126–133 = Mathematical Papers, Vol. X, pp. 323–330.Google Scholar
  19. [7]
    Cayley (A)., On the analytical forms called trees, American Journal of Mathematics, 4, 1881, pp. 266–268 = Mathematical Papers, Vol. XI, pp. 365–367.Google Scholar
  20. [8]
    Cayley (A)., On the theory of groups, American Journal of Mathematics, 11, 1889, pp. 139–157 = Mathematical Papers, Vol. XII, pp. 639–656.Google Scholar
  21. [9]
    Cayley (A)., A theorem on trees. Quarterly Journal of Mathematics, 23, 1889, pp. 376–378 = Mathematical Papers, Vol. XIII, pp. 26–28.Google Scholar
  22. [1]
    Churd (J.), Questions d’Analysis Situs (Thèse, Lausanne 1921,) Rendiconti del Circolo Matematico di Palermo, 46, 1922, pp. 185–224.CrossRefGoogle Scholar
  23. Churd (J.), (see Dumas).Google Scholar
  24. [1]
    Clausen (Th.), (Without title) Astronomische Nachrichten. 21, 1844, p, 216CrossRefGoogle Scholar
  25. [l]
    Dehn (M.), and Heegaard (P), Analysis Situs,Encyklopadie der mathematischen Wissenschaften, Vol. III 11, pp. 153–220, 1907.Google Scholar
  26. [l]
    Dumas (G.), and Chuard (J.), Sur les homologies de Poincaré, Comptes Rendus, Paris, 171, 1920, pp. 1113–1116.MATHGoogle Scholar
  27. [1]
    Errera (A.), Du coloriage des cartes et de quelques questions d’Analysis Situs, Thèse, Bruxelles, 1921.Google Scholar
  28. [2]
    Errera (A.), Une démonstration du théorème de Petersen, Mathesis, 36, 1922, pp. 56–61.MATHGoogle Scholar
  29. [3]
    Errera (A.), Un théorème sur les liaisons, Comptes Rendus, Paris, 177, 1923, pp. 489–491.MATHGoogle Scholar
  30. [4]
    Errera (A.), Exposé historique du problème des quatres couleurs, Periodico di Matematiche (4,) 7, 1927, pp. 20–41.Google Scholar
  31. [1]
    Euler (L.), Solutio problematis ad geometriam situs pertinentis, Commentarii Academiae Petropolitanae, 8, 1736 (1741) pp. 128–140 = Opera omnia, Ser. I, Vol. 7, pp. 1–10. French translation by Coupy in Nouvelles Annales de Mathématiques, 10, 1851, pp. 106- 118. German translation in Speiser: Klassische Stücke der Mathematik, (ürich 1927, pp. 127–138.Google Scholar
  32. [1]
    Frink (Orrin, Jr.), A proof of Petersen’s theorem. Annals of Mathematics (2,) 27, 1926, pp. 491–493.Google Scholar
  33. [1]
    Hajos (G.), Uum Mengerschen Graphensat,) Acta Litterarum ac Scientiarum (Sectio Scientiarum Mathematicarum,) Szeged, 7, 1934, pp. 44–47.MATHGoogle Scholar
  34. [1]
    Heawood (P.J.), Map-colour theorem. Quarterly Journal of Mathematics, 24, 1890, pp. 332–338 and 29, 1898, pp. 270–285.Google Scholar
  35. Heegaard (P.), (see Dehn).Google Scholar
  36. [1]
    Heffter (L.), Über das Problem der Nachbargebiete, Mathematische Annalen, 38, 1891, pp. 477–508.MathSciNetMATHGoogle Scholar
  37. [l]
    Hert (P.), Über Axiomensysteme für beliebige Satzsysteme, Mathematische Annalen, 87, 1922, pp. 246–269.MathSciNetCrossRefGoogle Scholar
  38. [1]
    Hierholer (C.), Ueber die Möglichkeit, einen Linienug ohne Wiederholung und ohne Unterbrechung zu umfahren. Mathematische Annalen, 6, 1873, pp. 30–32.Google Scholar
  39. [1]
    Hilton (H.), An introduction to the theory of groups of finite order, Oxford 1908.MATHGoogle Scholar
  40. [1]
    Jordan (C.), Sur les assemblages de lignes. Journal für die reine und angewandte Mathematik, 70, 1869, pp. 185–190.MATHCrossRefGoogle Scholar
  41. [1]
    Kempe (A.B.), On the geographical problem of the four colours, American Journal of Mathematics, 2, 1879, pp. 193–200.MathSciNetCrossRefGoogle Scholar
  42. [1]
    Kirchhoff (G.), Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Ströme geführt wird. Annalen der Physik und Chemie, 72, 1847, pp. 497–508 = Gesammelte Abhandlungen, Leipzig1882, pp. 22–33.Google Scholar
  43. [1]
    König (D.), Vonalrendszerek Kétoldalü felületeken (Graphs on two-sided surfaces,) Mathematikai és Természettudoményl Értesitö, 29, 1911, pp. 112–117 (Hungarian).Google Scholar
  44. [2]
    König (D.), A vonalrends(erek nems(amaról (Concerning the genus number of graphs,) Mathematikai és Természettudomanyi Értesitö, 29, 1911, pp. 345–350 (Hungarian).Google Scholar
  45. [3]
    König (D.), Sur un probième de la théorie générale des ensembles et la théorie des graphes (Communication made Apr. 7, 1914 in Paris to the Congrés de Philosophie Mathématique,) Revue de Métaphysique et de Morale, 30, 1923, pp. 443–449.Google Scholar
  46. [4]
    König (D.), Vonalrendszerek és determinénsok (Graphs and determinants,) Mathematikai és Természettudoményi Értesitö, 33, 1915, pp. 221–229 (Hungarian)Google Scholar
  47. [5]
    König (D.), Über Graphen und ihre Anwendung auf Determinanten- theorie und Mengenlehre, Mathematische Annalen, 77, 1916, pp. 453–465.MATHGoogle Scholar
  48. [6]
    König (D.), Sur les rapports topoiogiques d’un probième d’analyse combinatoire. Acta Litterarum ac Scientiarum (Sectio Scientiarum Mathematicarum,) Szeged, 2, 1924, pp. 32–38.Google Scholar
  49. [7]
    König (D.), Sur les correspondances multivoques des ensembles, Fundamenta Mathematicae, 8, 1926, pp. 114–134.MATHGoogle Scholar
  50. [8]
    König (D.), Über eine Schlueweise aus dem Endlichen ins Unendliche (Punktmengen. - Kartenfärben. - Verwandtschaftsbe(iehungen. - Schachspiel,) Acta Litterarum ac Seientiarum (Sectio Scientiarum Mathematicarum,) Szeged, 3, 1927, pp. 121–130.Google Scholar
  51. [9]
    König (D.), Graphok és matrixok (Graphs and matrices,) Matematikai és Fizikai Lapok, 38, 1931, pp. 116–119 (Hungarian with a German abstract)MATHGoogle Scholar
  52. [10]
    König (D.), Egy végességi tétel és alkalmazasai (A finiteness theorem with two applications,) Matematikai és Fizikai Lapok, 39, 1932, pp. 27–29 (Hungarian with a German abstract).Google Scholar
  53. [11]
    König (D.), Über trennende Knotenpunkte in Graphen znebst Anwendungen auf Determinanten und Matrizen,) Acta Litterarum ac Scientiarum (Sectio Scientiarum Mathematicarum,) Szeged, 6, 1933, pp. 155–179.MATHGoogle Scholar
  54. [1]
    König (D.), und Valkó (St.), Über mehrdeutige Abbildungen von Mengen, Mathematische Annalen, 95, 1926, pp. 135–138.Google Scholar
  55. [1]
    Kowalewski (A.), W.R. Hamiltons Dodekaederaufgabe als Buntordnungsproblem, Sit(ungsberichte der Akademie in Wien, 126, 1917, pp. 67–90.MATHGoogle Scholar
  56. [2]
    Kowalewski (A.), Topologische Deutung von Buntordnungsproblemen, Sit(ungsberichte der Akademie in Wien, 126, 1917, pp. 963–1007.MATHGoogle Scholar
  57. [1]
    Kowalewski (G.), Mathematica delectans, I: Boss-Puzzle und verwandte Spiele, Leipzig 1921.MATHGoogle Scholar
  58. [2]
    Kowalewski (G.), Alte und neue mathematische Spiele, Leipzig 1930.MATHGoogle Scholar
  59. [1]
    Kurschak (J.), Lóugras a végtelen sakktablan (The knight’s moves on an infinite chess board,) Mathematikai és Physikai Lapok, 33, 1926, pp. 117–119 (Hungarian)Google Scholar
  60. [2]
    Kurschak (J.), Rösselsprung auf dem unendlichen Schachbrette, Acta Litterarum ac Scientiarum (Sectio Scientiarum Mathematicarum,) Szeged, 4, 1928, pp. 12–13.Google Scholar
  61. [1]
    Kuratowski (C.), Sur le problème des courbes gauches en Topologie, Fundamenta Mathematicae, 15, 1930, pp. 271–283.MATHGoogle Scholar
  62. [1]
    Kuratowski (C.), et Whyburn (G.T.), Sur les éléments cycliques et leurs applications, Fundamenta Mathematicae, 16, 1930, pp. 305–331.MATHGoogle Scholar
  63. [1]
    Listing (J.B.), Vorstudien zur Topologie, Göttingen Studien, 1847; auch separat erschienen: Göttingen, 1848.Google Scholar
  64. [2]
    Listing (J.B.), Der Census räumlicher Complexe oder Verallgemeinerung des Eulerschen Satzes von den Polyedern, Göttinger Abhandlungen, 10, 1862.Google Scholar
  65. [1]
    Locas (E.), Récréations Mathématiques, I-IV, Paris 1882–1894.Google Scholar
  66. [2]
    Locas (E.), Théorie des nombres, I, Paris 1891.Google Scholar
  67. [1]
    Maschke (H.), The representation of finite groups… by Cayley’s color diagrams, American Journal of Mathematics, 18, 1896, pp. 156–194.MathSciNetMATHCrossRefGoogle Scholar
  68. [1]
    Menger (K.), Zur allgemeinen Kurventheorie, Fundamenta Mathematicae, 10, 1927, pp. 96–115.MATHGoogle Scholar
  69. [2]
    Menger (K.), Über reguläre Baumkurven, Mathematische Annalen, 96, 1927, pp. 572–582.MathSciNetCrossRefGoogle Scholar
  70. [3]
    Menger (K.), Über plättbare Dreiergraphen und Potenzen nichtplättbarer Graphen, Anzeiger der Akademie der Wissenschaften in Wien, 67, 1930, pp. 85–86 und Ergebnisse eines Mathematischen Kolloquiums, Heft 2, 1930, pp. 30–31.Google Scholar
  71. [4]
    Menger (K.), Kurventheorie (unter Mitarbeit von G. Nobeling,) Leipzig und Berlin 1932.Google Scholar
  72. [1]
    Petersen (J.), Die Theorie der regulären Graphen, Acta Mathematica, 15, 1891, pp. 193–220.MathSciNetMATHCrossRefGoogle Scholar
  73. [2]
    Petersen (J.), Sur le théorème de Tait, L’Intermédiaire des Mathématiciens, 5, 1898, pp. 225–227.Google Scholar
  74. [3]
    Petersen (J.), Les 36 officiers, Annuaire des mathématiciens 1901–02, Paris 1902, pp. 413- 427.Google Scholar
  75. [1]
    Poincaré (H.), (Premierz complémemt è 1’Analysis Situs, Rendiconti del Circolo Matematico di Palermo, 13, 1899, pp. 285–343.MATHCrossRefGoogle Scholar
  76. [2]
    Poincaré (H.), Second complément a 1’Analysis Situs, Proceedings of the London Mathematical Society, 32, 1901, pp. 277–308.CrossRefGoogle Scholar
  77. [1]
    de Polignac (C.), Formules et considérations diverses se rapportant a la théorie des amifications. Bulletin de la Société Mathématique de France, 8, 1880, pp. 120–124 and 9, 1881, pp. 30–42.Google Scholar
  78. [2]
    de Polignac (C.), Sur le théorème de Tait, Bulletin de la Société Mathématique de France, 27, 1899, pp. 142–145.MATHGoogle Scholar
  79. [1]
    Polya (G.), Lösung der Aufgabe 386 (von J. Schur,) Archiv der Mathematik und Physik (3,) 24, 1916, pp. 369–375.Google Scholar
  80. [1]
    Prufer (H.), Neuer Beweis eines Satzes über Permutationen, Archiv der Mathematik und Physik (3) 27, 1918, pp. 142–144.Google Scholar
  81. Rademacher (H.), (see Steinitz).Google Scholar
  82. [1]
    Rademacher (H.), und Toeplit (O.), Von Zahlen und Figuren, Berlin 1930MATHGoogle Scholar
  83. [1]
    Redei (L.), Ein Kombinatorischer Sat,) Acta Litterarum ac Scientiarum (Sectio Scientiarum MathematiCarum,) Szeged, 7, 1934, pp. 39–43MATHGoogle Scholar
  84. [1]
    Reimeisster (K.), Einführung in die kombinatorische Topologie (Die Wissenschaft, Vol. 86,) Braunschweig 1932.Google Scholar
  85. [1]
    Reynolds (C.N.), On the problem of coloring maps in four colors. Annals of Mathematics (2,) 28, 1927, pp. 1–15 and 477–492.MathSciNetGoogle Scholar
  86. [1]
    Rutt (N.E.), Concerning the cut points of a continuous curve when the arc curve AB contains exactly N independent arcs, American Journal of Mathematics, 51, 1929, pp. 217–246.MathSciNetMATHCrossRefGoogle Scholar
  87. [1]
    Saite-Lague (A.), Les réseaux, Comptes Rendus, Paris, 176, 1923, pp. 1202–1205.Google Scholar
  88. [2]
    Saite-Lague (A.), Les réseaux, Toulouse 1924Google Scholar
  89. [3]
    Saite-Lague (A.), Les réseaux unicursaux et bicursaux, Comptes Rendus, Paris, 182, 1926, pp. 747- 750.Google Scholar
  90. [4]
    Saite-Lague (A.), Les réseaux (ou graphes,) Mémorial des Sciences Mathématiques, Fase. 18, Paris 1926.Google Scholar
  91. [5]
    Saite-Lague (A.), Géométrie de situation et jeux, Mémoria! des Sciences Mathématiques, Fase. 41, Paris 1929.Google Scholar
  92. [1]
    Saite-Lague (A.), Ein Beweis des Petersenschen Graphensatzes, Acta Litterarum ac Seientiarum (Sectio Scientiarum MathematiCarum,) Szeged, 7, 1934, pp. 51–57.Google Scholar
  93. [1]
    Skolem (Th.), Gruppierungen, kombinatorische Reziprozitäten, Paarsysteme. Nachtrag (Kap. 15) zu Netto: Lehrbuch der Kombinatorik, 2nd ed., Leipzig and Berlin 1927.Google Scholar
  94. [1]
    v. Staudt (G.K.), Chr., Geometrie der Lage, Nürmberg 1847.Google Scholar
  95. Steinitz (E.), [1] Vorlesungen über die Theorie der Polyeder (unter Einschlue der Elemente der Topologie,) aus dem Nachlas herausgegeben und ergänzt von H. Rademacher (Grundlehren der math. Wiss., Vol. 41,) Berlin 1934.Google Scholar
  96. [1]
    Sylvester (J.J.), On recent discoveries in mechanical conversion of motion. Proceedings of the Royal Institution of Great Britain, 7, 1873–1875, pp. 179–198 = Mathematical Papers, Cambridge 1904–1912, vol. Ill, pp. 7–25.Google Scholar
  97. [2]
    Sylvester (J.J.), Problem 5208, Mathematical Questions with their Solutions from the Educational Times, 27, 1877; printed in Cayley’s Mathematical Papers, Vol. X, p. 598.Google Scholar
  98. [3]
    Sylvester (J.J.), On an application of the new atomic theory to the graphical representation of the invariants and covariants of binary quantics, American Journal of Mathematics, 1, 1878, pp. 64–125 = Mathematical Papers, vol. Ill, pp. 148–206.MathSciNetCrossRefGoogle Scholar
  99. [4]
    Sylvester (J.J.), On the geometrical forms called trees, Johns Hopkins University Circulars, I, 1882, pp. 202–203 = Mathematical Papers, vol. Ill, pp. 640–641.Google Scholar
  100. [1]
    Tait (P.G.), Note on a theorem in geometry of position. Transactions of the Royal Society of Edinbourgh, 29, 1880, pp. 657–660 = Scientific Papers, Cambridge 1898–1900, vol. I, pp. 408–411.Google Scholar
  101. [2]
    Tait (P.G.), Listing’s Topologie, Philosophical Magazine (5,) 17, 1884, pp. 30–46 = Scientific Papers, Vol. II, pp. 85–98Google Scholar
  102. [1]
    Tarry (G.), Le problème des labyrinthes, Nouvelles Annales de Mathématiques (3,) 14, 1895, pp. 187–190.Google Scholar
  103. [1]
    Terquem (O.), Sur les polygones et les polyèdres étoilés, polygones funiculaires (d’après M. Poinsot,) Nouvelles Annales de Mathématiques, 8, 1849, pp. 68–74Google Scholar
  104. Toeplitz (O.), (see Rademacher).Google Scholar
  105. Ulam (S.), Remark on the generalised Bernstein’s theorem, Fundamenta Mathematicae, 13, 1929, pp. 281–283.MATHGoogle Scholar
  106. Valko (St.), (see König).Google Scholar
  107. [1]
    Veblen (O.), An application of modular equations in Analysis Situs, Annals of Mathematics (2,) 14, 1912, pp. 86–94.MathSciNetCrossRefGoogle Scholar
  108. [2]
    Veblen (O.), Analysis Situs, The Cambridge Colloquium 1916, part II, New York, 1st edition 1922, 2nd edition 1931.Google Scholar
  109. [1]
    Veblen (O.), and Alexander (J.W.), Manifolds of N dimensions. Annals of Mathematics (2,) 14, 1913, pp. 163–178.MathSciNetGoogle Scholar
  110. [1]
    van der Waerden (B.L.), Ein Satz über Klasseneinteilungen von endlichen Mengen, Abhandlungen aus dem mathematischen Seminar der Hamburgisehen Universität, 5, 1927, pp. 185–188MATHCrossRefGoogle Scholar
  111. [1]
    Weyl (H.), Repartición de corriente en una red conductora (Introducción al Analisis situs combinatorio,) Revista Matemética Hispano- Americana, 5, 1923, pp. 153–164.Google Scholar
  112. [1]
    Whitney (H.), A theorem on graphs. Annals of Mathematics (2). 32, 1931, pp. 378–390.MathSciNetCrossRefGoogle Scholar
  113. [2]
    Whitney (H.), Congruent graphs and connectivity of graphs, American Journal of Mathematics, 54, 1932, pp. 150–168.MathSciNetCrossRefGoogle Scholar
  114. [3]
    Whitney (H.), Non-separable and planar graphs. Transactions of the American Mathematical Society, 34, 1932, pp. 339–362 (an abstract appeared in Proceedings of the National Academy of Sciences of the U.S.A., 17, 1931, pp. 125–127).MathSciNetCrossRefGoogle Scholar
  115. [4]
    Whitney (H.), Planar Graphs, Fundamenta Mathematicae, 21, 1933, pp. 73–84.Google Scholar
  116. [1]
    Whyburn (G.T.), Cyclicly connected continuous curves. Proceedings of the National Academy of Sciences of the U.S.A., 13, 1927, pp. 31–38.MATHCrossRefGoogle Scholar
  117. Whyburn (G.T.), (see Kuratowski).Google Scholar
  118. [1]
    Wiener (Chr.), Ueber eine Aufgabe aus der Geometria situs. Mathematische Annalen, 6, 1873, pp. 29–30.MathSciNetMATHGoogle Scholar

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  • Dénes König

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