Covering, Coloring, and Packing Hypergraphs

  • Rudolf AhlswedeEmail author
Part of the Foundations in Signal Processing, Communications and Networking book series (SIGNAL, volume 13)


A hypergraph  \(\mathcal {H}=(\mathcal V,\mathcal E)\) consists of a (finite) vertex set \(\mathcal V\) and a set of hyper-edges \(\mathcal E\), where each edge \(E\in \mathcal E\) is a subset of \(E\subset \mathcal V\). The vertices will usually be labelled by \(\mathcal V=(v_1,\dots ,v_I)\), the edges by \(\mathcal E=(E_1,\dots ,E_J)\), where \(I,J\in {\mathbb N}\) with \(I=|\mathcal V|\) and \(1\le J\le 2^{|\mathcal E|}\).


Coloring Lemmas Orthogonal Color Strict Colorings Slepian-Wolf Theorem Hyper Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    A. Ahlswede, I. Althöfer, C. Deppe, U. Tamm (eds.), Storing and Transmitting Data, Rudolf Ahlswede’s Lectures on Information Theory 1, Foundations in Signal Processing, Communications and Networking, vol. 10, 1st edn. (Springer, 2014)Google Scholar
  2. 2.
    R. Ahlswede, V. Blinovsky, Lectures on Advances in Combinatorics (Springer, Berlin, 2008)zbMATHCrossRefGoogle Scholar
  3. 3.
    G. Birkhoff, Three observations on linear algebra. Univ. Nac. Tucumán. Revista A. 5, 147–151 (1946)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Z. Blázsik, M. Hujter, A. Pluhaár, Z. Tuza, Graphs with no induced C4 and 2K2. Discret. Math. 115, 51–55 (1993)CrossRefGoogle Scholar
  5. 5.
    T.M. Cover, J.A. Thomas, Elements of Information Theory, 2nd edn. (Wiley, New York, 2006)zbMATHGoogle Scholar
  6. 6.
    H. Hadwiger, Über eine Klassifikation der Streckenkomplexe. Vierteljschr. Naturforsch. Ges. Zürich 88, 133–143 (1943)MathSciNetzbMATHGoogle Scholar
  7. 7.
    G.G. Lorentz, On a problem of additive number theory. Proc. Am. Math. Soc. 5(5), 838–841 (1954)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    L. Lóvasz, Minimax theorems for hypergraphs, in Hypergraph Seminar. Lecture Notes in Mathematics, vol. 441 (Springer, Berlin, 1974), pp. 111–126Google Scholar
  9. 9.
    L. Lóvasz, On the ratio of optimal integral and fractional covers. Discret. Math. 13, 383–390 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    V. Rödl, On a packing and covering problem. Eur. J. Comb. 5, 69–78 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    M. Rosenfeld, On a problem of C.E. Shannon in graph theory. Proc. Am. Math. Soc. 18, 315–319 (1967)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    C.E. Shannon, The zero error capacity of a noisy channel. I.R.E. Trans. Inf. Theory IT–2, 8–19 (1956)Google Scholar
  13. 13.
    S. Stahl, \(n\)-tuple colorings and associated graphs. J. Comb. Theory (B) 29, 185–203 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    V.G. Vizing, A bound on the external stability number of a graph. Dokl. Akad. Nauk SSSR 164, 729–731 (1965)MathSciNetGoogle Scholar

Further Readings

  1. 15.
    M.O. Albertson, J.P. Hutchinson, On six-chromatic toroidal graphs. Proc. Lond. Math. Soc. 3(41), 533–556 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 16.
    R. Aharoni, I. Ben-Arroyo, A.J.Hoffman Hartman, Path-partitions and packs of acyclic digraphs. Pacific J. Math. 118, 249–259 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 17.
    S. Benzer, On the topology of the genetic fine structure. Proc. National Acad. Sci. U. S. A. 45(11), 1607–1620 (1959)Google Scholar
  4. 18.
    C. Berge, Théorie des graphes et ses applications (Dunod, Paris, 1958)zbMATHGoogle Scholar
  5. 19.
    C. Berge, Les problémes de coloration en Théorie des Graphes. Publ. Inst. Statist. Univ. Paris 9, 123–160 (1960)MathSciNetzbMATHGoogle Scholar
  6. 20.
    C. Berge, Färbung von Graphen, deren sämtliche bzw. deren ungerade Kreise starr sind, Wiss. Zeitschrift der Martin-Luther-Universität Halle-Wittenberg, 114–115 (1961)Google Scholar
  7. 21.
    C. Berge, The Theory of Graphs and its Applications (Methuen, London, 1961), p. 95Google Scholar
  8. 22.
    C. Berge, Sur un conjecture relative au probleme des codes optimaux, Comm. 13ieme Assemblee Gen. URSI, Tokyo (1962)Google Scholar
  9. 23.
    C. Berge, Perfect graphs, in Six Papers on Graph Theory (Indian Statistical Institute, Calcutta, Research and Training School, 1963), pp. 1–21Google Scholar
  10. 24.
    C. Berge, Une Application de la Théorie des Graphes à un problème de Codage, in Automata Theory, ed. by E.R. Caianiello (Academic Press, New York, 1966), pp. 25–34Google Scholar
  11. 25.
    C. Berge, Some classes of perfect graphs, in Graph Theory and Theoretical Physics (Academic Press, New York, 1967), pp. 155–165Google Scholar
  12. 26.
    C. Berge, The rank of a family of sets and some applications to graph theory, in Recent Progress in Combinatorics (Proceedings of the Third Waterloo Conference on Combinatorics, 1968) (Academic Press, New York, 196), pp. 49–57Google Scholar
  13. 27.
    C. Berge, Some classes of perfect graphs, in Combinatorial Mathematics and its Applications, Proceedings of the Conference Held at the University of North Carolina, Chapel Hill, 1967 (University of North Carolina Press, 539–552, 1969)Google Scholar
  14. 28.
    C. Berge, Graphes et Hypergraphes, Monographies Universitaires de Mathématiques, No. 37. Dunod, Paris (1970)Google Scholar
  15. 29.
    C. Berge, Balanced matrices. Math. Program. 2(1), 19–31 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 30.
    C. Berge, Graphs and Hypergraphs, Translated from the French by Edward Minieka, North-Holland Mathematical Library, vol. 6. North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York (1973)Google Scholar
  17. 31.
    C. Berge, A theorem related to the Chvátal conjecture, in Proceedings of the Fifth British Combinatorial Conference (University of Aberdeen, Aberdeen, 1975), Congressus Numerantium, No. XV, Utilitas Math., Winnipeg, Man. (1976), pp. 35–40Google Scholar
  18. 32.
    C. Berge, \(k\)-optimal partitions of a directed graph. Eur. J. Comb. 3, 97–101 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 33.
    C. Berge, Path-partitions in directed graphs, in Combinatorial Mathematics, ed. by C. Berge, D. Bresson, P. Camion, J.F. Maurras, F. Sterboul (North-Holland, Amsterdam, 1983), pp. 32–44Google Scholar
  20. 34.
    C. Berge, A property of k-optimal path-partitions, in Progress in Graph Theory, ed. by J.A. Bondy, U.S.R. Murty (Academic Press, New York, 1984), pp. 105–108Google Scholar
  21. 35.
    C. Berge, On the chromatic index of a linear hypergraph and the Chvátal conjecture, in Annals of the New York Academy of Sciences, vol. 555, ed. by G.S. Bloom, R.L. Graham, J. Malkevitch, C. Berge (1989), pp. 40–44Google Scholar
  22. 36.
    C. Berge, Hypergraphs, Combinatorics of Finite Sets, Chapter 1, Section 4 (North-Holland, New York, 1989)Google Scholar
  23. 37.
    C. Berge, On two conjectures to generalize Vizing’s Theorem. Le Matematiche 45, 15–24 (1990)MathSciNetzbMATHGoogle Scholar
  24. 38.
    C. Berge, The \(q\)-perfect graphs I: the case \(q = 2\), in Sets, Graphs and Numbers, ed. by L. Lovász, D. Miklós, T. Szönyi. Colloq. Math. Soc. Janós Bolyai, vol. 60 (1992), pp. 67–76Google Scholar
  25. 39.
    C. Berge, The \(q\)-perfect graphs II, in Graph Theory, Combinatorics and Applications, ed. by Y. Alavi, A. Schwenk (Wiley Interscience, New York, 1995), pp. 47–62Google Scholar
  26. 40.
    C. Berge, The history of the perfect graphs. Southeast Asian Bull. Math. 20(1), 5–10 (1996)MathSciNetzbMATHGoogle Scholar
  27. 41.
    C. Berge, Motivations and history of some of my conjectures, in Graphs and combinatorics (Marseille, 1995) (1997), pp. 61–70 (Discrete Math. 165–166)Google Scholar
  28. 42.
    C. Berge, V. Chvátal (eds.), Topics on Perfect Graphs. Annals of Discrete Mathematics, vol. 21 (North Holland, Amsterdam, 1984)Google Scholar
  29. 43.
    C. Berge, P. Duchet, Strongly perfect graphs, in Topics on Perfect Graphs, ed. by C. Berge, V. Chvátal. North-Holland Mathematics Studies, vol. 88 (North-Holland, Amsterdam, 1984), pp. 57–61 (Annals of Disc. Math. 21)Google Scholar
  30. 44.
    C. Berge, A.J.W. Hilton, On two conjectures about edge colouring for hypergraphs. Congr. Numer. 70, 99–104 (1990)MathSciNetzbMATHGoogle Scholar
  31. 45.
    C. Berge, M. Las, Vergnas, Sur un théorème du type König pour hypergraphes. Ann. New York Acad. Sci. 175, 32–40 (1970)MathSciNetzbMATHGoogle Scholar
  32. 46.
    I. Ben-Arroyo Hartman, F. Sale, D. Hershkowitz, On Greene’s Theorem for digraphs. J. Graph Theory 18, 169–175 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 47.
    A. Beutelspacher, P.-R. Hering, Minimal graphs for which the chromatic number equals the maximal degree. Ars Combinatoria 18, 201–216 (1983)MathSciNetzbMATHGoogle Scholar
  34. 48.
    O.V. Borodin, A.V. Kostochka, An upper bound of the graph’s chromatic number, depending on the graph’s degree and density. J. Comb. Theory B 23, 247–250 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 49.
    A. Brandstädt, V.B. Le, J.P. Spinrad, Graph classes: a survey (SIAM Monographs on Discrete Mathematics and Applications (SIAM, Philadelphia, 1999)zbMATHCrossRefGoogle Scholar
  36. 50.
    R.C. Brigham, R.D. Dutton, A compilation of relations between graph invariants. Networks 15(1), 73–107 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 51.
    R.C. Brigham, R.D. Dutton, A compilation of relations between graph invariants: supplement I. Networks 21, 412–455 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 52.
    R.L. Brooks, On colouring the nodes of a network. Proc. Camb. Philos. Soc. 37, 194–197 (1941)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 53.
    M. Burlet, J. Fonlupt, Polynomial algorithm to recognize a Meyniel graph, in Topics on Perfect Graphs, ed. by C. Berge, V. Chvátal. North-Holland Mathematics Studies, vol. 88 (North-Holland, Amsterdam, 1984), pp. 225–252 (Annals of Discrete Math. 21)Google Scholar
  40. 54.
    K. Cameron, On k-optimum dipath partitions and partial k-colourings of acyclic digraphs. Eur. J. Comb. 7, 115–118 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 55.
    P.J. Cameron, A.G. Chetwynd, J.J. Watkins, Decomposition of snarks. J. Graph Theory 11, 13–19 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 56.
    P. Camion, Matrices totalement unimodulaires et problèmes combinatoires (Université Libre de Bruxelles, Thèse, 1963)Google Scholar
  43. 57.
    P.A. Catlin, Another bound on the chromatic number of a graph. Discret. Math. 24, 1–6 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 58.
    W.I. Chang, E. Lawler, Edge coloring of hypergraphs and a conjecture of Erdös-Faber-Lovász. Combinatorica 8, 293–295 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 59.
    C.-Y. Chao, On a problem of C. Berge. Proc. Am. Math. Soc. 14, 80 (1963)zbMATHCrossRefGoogle Scholar
  46. 60.
    M. Chudnovsky, G. Cornuejols, X. Liu, P. Seymour, K. Vuskovic, Recognizing Berge graphs. Combinatorica 25, 143–186 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 61.
    M. Chudnovsky, N. Robertson, P. Seymour, R. Thomas, The strong perfect graph theorem. Ann. Math. 164, 51–229 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 62.
    V. Chvátal, Unsolved problem no. 7, in Hypergraph Seminar, ed. by C. Berge, D.K. Ray-Chaudhuri. Lecture Notes in Mathematics, vol. 411 (Springer, Berlin, 1974)Google Scholar
  49. 63.
    V. Chvátal, Intersecting families of edges in hypergraphs having the hereditary property, in Hypergraph Seminar (Proceedings of the First Working Seminar, Ohio State University, Columbus, Ohio, 1972; dedicated to Arnold Ross). Lecture Notes in Mathematics, vol. 411 (Springer, Berlin, 1974), pp. 61–66Google Scholar
  50. 64.
    V. Chvátal, On certain polytopes associated with graphs. J. Comb. Theory Ser. B 18, 138–154 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 65.
    V. Chvátal, On the strong perfect graph conjecture. J. Comb. Theory Ser. B 20, 139–141 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 66.
    V. Chvátal, Perfectly ordered graphs, in Topics on Perfect Graphs, ed. by C. Berge, V. Chvátal. North-Holland Mathematics Studies, vol. 88 (North-Holland, Amsterdam, New York, 1984), pp. 63–65 (Annals of Disc. Math. 21)Google Scholar
  53. 67.
    V. Chvátal, Star-cutsets and perfect graphs. J. Comb. Theory Ser. B 39(3), 189–199 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 68.
    V. Chvátal, J. Fonlupt, L. Sun, A. Zemirline, Recognizing dart-free perfect graphs. SIAM J. Comput. 31(5), 1315–1338 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 69.
    V. Chvátal, D.A. Klarner, D.E. Knuth, Selected combinatorial research problems, Technical report STAN-CS, 72-292 (1972)Google Scholar
  56. 70.
    J. Colbourn, M. Colbourn, The chromatic index of cyclic Steiner 2-design. Int. J. Math. Sci. 5, 823–825 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 71.
    M. Conforti, G. Cornuéjols, Graphs without odd holes, parachutes or proper wheels: a generalization of Meyniel graphs and of line graphs of bipartite graphs. J. Comb. Theory Ser. B 87, 300–330 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 72.
    M. Conforti, M.R. Rao, Structural properties and decomposition of linear balanced matrices. Math. Program. Ser. A B 55(2), 129–168 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 73.
    M. Conforti, M.R. Rao, Articulation sets in linear perfect matrices I: forbidden configurations and star cutsets. Discret. Math. 104(1), 23–47 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 74.
    M. Conforti, M.R. Rao, Articulation sets in linear perfect matrices II: the wheel theorem and clique articulations. Discret. Math. 110(1–3), 81–118 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 75.
    M. Conforti, M.R. Rao, Testing balancedness and perfection of linear matrices. Math. Program. Ser. A 61(1), 1–18 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 76.
    M. Conforti, G. Cornuéjols, A. Kapoor, K. Vuskovic, A mickey-mouse decomposition theorem, in Integer Programming and Combinatorial Optimization (Copenhagen, 1995). Lecture Notes in Computer Science, vol. 920 (Springer, Berlin, 1995), pp. 321–328Google Scholar
  63. 77.
    M. Conforti, G. Cornuéjols, A. Kapoor, K. Vuskovic, Even and odd holes in cap-free graphs. J. Graph Theory 30(4), 289–308 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 78.
    M. Conforti, G. Cornuéjols, M.R. Rao, Decomposition of balanced matrices. J. Comb. Theory Ser. B 77(2), 292–406 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  65. 79.
    M. Conforti, B. Gerards, A. Kapoor, A theorem of Truemper. Combinatorica 20(1), 15–26 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  66. 80.
    M. Conforti, G. Cornuéjols, A. Kapoor, K. Vuskovic, Balanced \(0, {+\atop -}1\) matrices I, decomposition. J. Comb. Theory Ser. B 81(2), 243–274 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 81.
    M. Conforti, G. Cornuéjols, A. Kapoor, K. Vuskovic, Balanced \(0, {+\atop -}1\) matrices II, recognition algorithm. J. Comb. Theory Ser. B 81(2), 275–306 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  68. 82.
    M. Conforti, G. Cornuéjols, G. Gasparyan, K. Vuskovic, Perfect graphs, partitionable graphs and cutsets. Combinatorica 22(1), 19–33 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 83.
    M. Conforti, G. Cornuéjols, A. Kapoor, K. Vuskovic, Even-hole free graphs, part I: decomposition theorem. J. Graph Theory 39(1), 6–49, vol. 40 (2002)Google Scholar
  70. 84.
    M. Conforti, G. Cornuéjols, A. Kapoor, K. Vuskovic, Even-hole free graphs, part II: recognition algorithm. J. Graph Theory 40(4), 238–266 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  71. 85.
    M. Conforti, G. Cornuéjols, K. Vuskovic, Decomposition of odd-hole-free graphs by double star cutsets and 2-joins. Discret. Appl. Math. 141(1–3), 41–91 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  72. 86.
    M. Conforti, G. Cornuéjols, K. Vuskovic, Square-free perfect graphs. J. Comb. Theory B 257–307 (2004)Google Scholar
  73. 87.
    G. Cornuéjols, Combinatorial optimization: packing and covering, in CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 74 (SIAM, Philadelphia, 2001)CrossRefGoogle Scholar
  74. 88.
    G. Cornuéjols, The strong perfect graph conjecture, in Proceedings of the International Congress of Mathematicians III: Invited Lectures Beijing (2002), pp. 547–559Google Scholar
  75. 89.
    G. Cornuéjols, W.H. Cunningham, Compositions for perfect graphs. Discret. Math. 55(3), 245–254 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  76. 90.
    G. Cornuéjols, B. Reed, Complete multi-partite cutsets in minimal imperfect graphs. J. Comb. Theory Ser. B 59(2), 191–198 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  77. 91.
    I. Csiszár, J. Körner, Information Theory. Coding Theorems for Discrete Memoryless Systems. Probability and Mathematical Statistics (Academic Press Inc., New York, 1981)Google Scholar
  78. 92.
    I. Csiszár, J. Körner, L. Lovász, K. Marton, G. Simonyi, Entropy splitting for antiblocking corners and perfect graphs. Combinatorica 10(1), 27–40 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  79. 93.
    W.H. Cunningham, J.A. Edmonds, A combinatorial decomposition theory. Can. J. Math. 32(3), 734–765 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  80. 94.
    C.M.H. de Figueiredo, S. Klein, Y. Kohayakawa, B. Reed, Finding skew partitions efficiently. J. Algorithms 37, 505–521 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  81. 95.
    B. Descartes, A three colour problem, Eureka (April 1947; solution March 1948) and Solution to Advanced Problem No. 4526, Amer. Math. Monthy, vol. 61 (1954), p. 352Google Scholar
  82. 96.
    R.P. Dilworth, A decomposition theorem for partially ordered sets. Ann. Math. 2, 161–166 (1950)MathSciNetzbMATHCrossRefGoogle Scholar
  83. 97.
    G.A. Dirac, Map-colour theorems. Can. J. Math. 4, 480–490 (1952)MathSciNetzbMATHCrossRefGoogle Scholar
  84. 98.
    G.A. Dirac, On rigid circuit graphs. Abh. Math. Sem. Univ. Hamburg 25, 71–76 (1961)MathSciNetzbMATHCrossRefGoogle Scholar
  85. 99.
    R.J. Duffin, The extremal length of a network. J. Math. Anal. Appl. 5, 200–215 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  86. 100.
    R.D. Dutton, R.C. Brigham, INGRID: a software tool for extremal graph theory research. Congr. Numerantium 39, 337–352 (1983)zbMATHGoogle Scholar
  87. 101.
    R.D. Dutton, R.C. Brigham, F. Gomez, INGRID: a graph invariant manipulator. J. Symb. Comput. 7, 163–177 (1989)zbMATHCrossRefGoogle Scholar
  88. 102.
    J. Edmonds, Minimum partition of a matroid into independent subsets. J. Res. Nat. Bur. Stand. Sect. B 69B, 67–72 (1965)Google Scholar
  89. 103.
    J. Edmonds, Maximum matching and a polyhedron with \(0,1\)-vertices. J. Res. Nat. Bur. Stand. Sect. B 69B, 125–130 (1965)Google Scholar
  90. 104.
    J. Edmonds, Paths, trees, and flowers. Can. J. Math. 17, 449–467 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  91. 105.
    J. Edmonds, Lehman’s switching game and a theorem of Tutte and Nash-Williams. J. Res. Nat. Bur. Stand. Sect. B 69B, 73–77 (1965)Google Scholar
  92. 106.
    J. Edmonds, Optimum branchings. J. Res. Nat. Bur. Stand. Sect. B 71B, 233–240 (1967)Google Scholar
  93. 107.
    J. Edmonds, Submodular functions, matroids, and certain polyhedra, in Combinatorial Structures and their Applications (Proceedings of the Calgary International Conference, Calgary, Alberta, 1969) (Gordon and Breach, New York, 1970), pp. 69–87Google Scholar
  94. 108.
    J. Edmonds, Matroids and the greedy algorithm, (Lecture, Princeton, 1967). Math. Programming 1, 127–136 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  95. 109.
    J. Edmonds, Edge-disjoint branchings, in Combinatorial Algorithms (Courant Computer Science Symposium 9, New York University, New York, 1972) (Algorithmics Press, New York, 1973), pp. 91–96Google Scholar
  96. 110.
    J. Edmonds, Submodular functions, matroids, and certain polyhedra, in Combinatorial optimization-Eureka, you shrink!. Lecture Notes in Computer Science, vol. 2570 (Springer, Berlin, 2003), pp. 11–26Google Scholar
  97. 111.
    J. Edmonds, D.R. Fulkerson, Bottleneck extrema. J. Comb. Theory 8, 299–306 (1970)CrossRefGoogle Scholar
  98. 112.
    P. Erdös, Graph theory and probability. Canad. J. Math. 11, 34–38 (1959)MathSciNetzbMATHCrossRefGoogle Scholar
  99. 113.
    P. Erdös, Problems and results in Graph Theory, in Proceedings of the 5th British Combinatorial Conference, ed. by C.St.J.A. Nash-Williams, J. Sheehan. Utilitas Math., vol. 15 (1976)Google Scholar
  100. 114.
    P. Erdös, A. Hajnal, On chromatic number of graphs and set-systems. Acta Math. Acad. Sci. Hung. 17, 61–99 (1966)MathSciNetzbMATHCrossRefGoogle Scholar
  101. 115.
    P. Erdös, V. Faber, L. Lovász, Open problem, in Hypergraph Seminar, ed. by C. Berge, D. Ray Chaudhuri. Lecture Notes in Mathematics, vol. 411 (Springer, Berlin, 1974)Google Scholar
  102. 116.
    P. Erdös, C. Ko, R. Rado, Intersection theorems for systems of finite sets. Quart. J. Math. Oxford Ser. 2(12), 313–320 (1961)MathSciNetzbMATHCrossRefGoogle Scholar
  103. 117.
    J. Fonlupt, J.P. Uhry, Transformations which preserve perfectness and \(H\)-perfectness of graphs. Ann. Discret. Math. 16, 83–95 (1982)MathSciNetzbMATHGoogle Scholar
  104. 118.
    J. Fonlupt, A. Zemirline, A polynomial recognition algorithm for perfect \(K_4-\{e\}\)-free graphs, Rapport Technique RT-16 (Artemis, IMAG, Grenoble, France, 1987)Google Scholar
  105. 119.
    J.-L. Fouquet, Perfect Graphs with no \(2K_2\) and no \(K_6\), Technical report, Universite du Maine, Le Mans, France (1999)Google Scholar
  106. 120.
    J.-L. Fouquet, F. Maire, I. Rusu, H. Thuillier, Unpublished internal report (Univ, Orléans, LIFO, 1996)Google Scholar
  107. 121.
    L.R. Ford, D.R. Fulkerson, Flows in Networks (Princeton University Press, Princeton, 1962)zbMATHGoogle Scholar
  108. 122.
    D.R. Fulkerson, The maximum number of disjoint permutations contained in a matrix of zeros and ones. Can. J. Math. 16, 729–735 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  109. 123.
    D.R. Fulkerson, Networks, frames, blocking systems, in Mathematics of the Decision Sciences, Part 1, (Seminar, Stanford, California, 1967) (American Mathematical Society, Providence, 1968), pp. 303–334Google Scholar
  110. 124.
    D.R. Fulkerson, The perfect graph conjecture and pluperfect graph theorem, in 2nd Chapel Hill Conference on Combinatorial Mathematics and its Applications, Chapel Hill, N.C. (1969), pp. 171–175Google Scholar
  111. 125.
    D.R. Fulkerson, Notes on combinatorial mathematics: anti-blocking polyhedra, Rand corporation, Memorandum RM-6201/1-PR (1970)Google Scholar
  112. 126.
    D.R. Fulkerson, Blocking polyhedra, in Graph theory and its Applications (Academic, New York, 1970), pp. 93–111Google Scholar
  113. 127.
    D.R. Fulkerson, Blocking and anti-blocking pairs of polyhedra. Math. Program. 1, 168–194 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  114. 128.
    D.R. Fulkerson, Disjoint common partial transversals of two families of sets, in Studies in Pure Mathematics (Presented to Richard Rado) (Academic Press, London, 1971), pp. 107–112Google Scholar
  115. 129.
    D.R. Fulkerson, Anti-blocking polyhedra. J. Comb. Theory Ser. B 12, 50–71 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  116. 130.
    D.R. Fulkerson, On the perfect graph theorem, in Mathematical Progamming (Proceedings of the Advanced Seminar, University of Wisconsin, Madison, Wisconsin, 1972), ed. by T.C. Hu, S.M. Robinson, Mathematical Research Center Publications, vol. 30, (Academic Press, New York, 1973), pp. 69–76Google Scholar
  117. 131.
    D.R. Fulkerson (ed.), Studies in Graph Theory. Studies in Mathematics, vol. 12 (The Mathematical Association of America, Providence, 1975)Google Scholar
  118. 132.
    Z. Füredi, The chromatic index of simple hypergraphs. Res. Problem Graphs Comb. 2, 89–92 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  119. 133.
    T. Gallai, Maximum-minimum Sätze über Graphen. Acta Math. Acad. Sci. Hungar. 9, 395–434 (1958)MathSciNetzbMATHCrossRefGoogle Scholar
  120. 134.
    T. Gallai, Über extreme Punkt- und Kantenmengen. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 2, 133–138 (1959)MathSciNetzbMATHGoogle Scholar
  121. 135.
    T. Gallai, Graphen mit triangulierbaren ungeraden Vielecken, Magyar Tud. Akad. Mat. Kutató Int. Közl. 7, 3–36 (1962)MathSciNetzbMATHGoogle Scholar
  122. 136.
    T. Gallai, On directed paths and circuits, in Theory of Graphs, ed. by P. Erdös, G. Katona (Academic Press, New York, 1968), pp. 115–118Google Scholar
  123. 137.
    T. Gallai, A.N. Milgram, Verallgemeinerung eines graphentheoretischen Satzes von Rédei. Acta Sci. Math. 21, 181–186 (1960)MathSciNetzbMATHGoogle Scholar
  124. 138.
    F. Gavril, Algorithms on circular-arc graphs. Networks 4, 357–369 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  125. 139.
    J.F. Geelen, Matchings, Matroids and unimodular Matrices, Ph.D. thesis, University of Waterloo, 1995Google Scholar
  126. 140.
    D. Gernert, A knowledge-based system for graph theory. Methods Oper. Res. 63, 457–464 (1989)Google Scholar
  127. 141.
    D. Gernert, Experimental results on the efficiency of rule-based systems, in Operations Research ’92, ed. by A. Karmann et al. (1993), pp. 262–264Google Scholar
  128. 142.
    D. Gernert, Cognitive aspects of very large knowledge-based systems. Cogn. Syst. 5, 113–122 (1999)Google Scholar
  129. 143.
    D. Gernert, L. Rabern, A knowledge-based system for graph theory, demonstrated by partial proofs for graph-colouring problems. MATCH Commun. Math. Comput. Chem. 58(2), 445–460 (2007)MathSciNetzbMATHGoogle Scholar
  130. 144.
    A. Ghouila-Houri, Sur une conjecture de Berge (mimeo.), Institut Henri Poincaré (1960)Google Scholar
  131. 145.
    A. Ghouila-Houri, Caractérisation des matrices totalement unimodulaires. C. R. Acad. Sci. Paris 254, 1192–1194 (1962)MathSciNetzbMATHGoogle Scholar
  132. 146.
    A. Ghouila-Houri, Caractérisation des graphes non orientés dont on peut orienter les arêtes de maniére à obtenir le graphe d’une relation d’ordre. C. R. Acad. Sci. Paris 254, 1370–1371 (1962)MathSciNetzbMATHGoogle Scholar
  133. 147.
    P.C. Gilmore, A.J. Hoffman, A characterization of comparability graphs and of interval graphs. Canad. J. Math. 16, 539–548 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  134. 148.
    M. Gionfriddo, Zs. Tuza, On conjectures of Berge and Chvátal. Discret. Math. 124, 76–86 (1994)CrossRefGoogle Scholar
  135. 149.
    M.K. Goldberg, Construction of class 2 graphs with maximum vertex degree 3. J. Comb. Theory Ser. B 31, 282–291 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  136. 150.
    M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs. Computer Science and Applied Mathematics (Academic Press, New York, 1980). Second edition, Annals of Discrete Mathematics 57, Elsevier, 2004Google Scholar
  137. 151.
    R. Gould, Graph Theory (Benjamin Publishing Company, Menlo Park, 1988)zbMATHGoogle Scholar
  138. 152.
    C. Greene, D.J. Kleitman, The structure of Sperner k-families. J. Comb. Theory Ser. A 34, 41–68 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  139. 153.
    M. Grötschel, L. Lovász, A. Schrijver, Geometric Algorithms and Combinatorial Optimization (Springer, Berlin, 1988)zbMATHCrossRefGoogle Scholar
  140. 154.
    A. Hajnal, J. Surányi, Über die Auflösung von Graphen in vollständige Teilgraphen. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 1, 53–57 (1958)zbMATHGoogle Scholar
  141. 155.
    G. Hajos, Über eine Art von Graphen, Int. Math. Nachr. 11 (1957)Google Scholar
  142. 156.
    F. Harary, C. Holtzmann, Line graphs of bipartite graphs. Rev. Soc. Mat. Chile 1, 19–22 (1974)Google Scholar
  143. 157.
    M. Henke, A. Wagler, Auf dem Weg von der Vermutung zum Theorem: Die Starke-Perfekte-Graphen-Vermutung. DMV-Mitteilungen 3, 22–25 (2002)MathSciNetGoogle Scholar
  144. 158.
    N. Hindman, On a conjecture of Erdös, Farber. Lovász about n-colorings. Canad. J. Math. 33, 563–570 (1981)zbMATHCrossRefGoogle Scholar
  145. 159.
    C.T. Hoàng, Some properties of minimal imperfect graphs. Discret. Math. 160(1–3), 165–175 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  146. 160.
    A.J. Hoffman, Some recent applications of the theory of linear inequalities to extremal combinatorial analysis. Proc. Sympos. Appl. Math. 10, 113–127 (1960)MathSciNetCrossRefGoogle Scholar
  147. 161.
    A.J. Hoffman, Extending Greene’s Theorem to directed graphs. J. Comb. Theory Ser. A 34, 102–107 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  148. 162.
    A.J. Hoffman, J.B. Kruskal, Integral boundary points of convex polyhedra, in Linear Inequalities and Related Systems. Annals of Mathematics Studies, vol. 38 (Princeton University Press, Princeton, 1956), 223–246Google Scholar
  149. 163.
    P. Horák, A coloring problem related to the Erdös-Faber-Lovász Conjecture. J. Comb. Theory Ser. B 50, 321–322 (1990)zbMATHCrossRefGoogle Scholar
  150. 164.
    S. Hougard, A. Wagler, Perfectness is an elusive graph property, Preprint ZR 02–11, ZIB, 2002. SIAM J. Comput. 34(1), 109–117 (2005)CrossRefGoogle Scholar
  151. 165.
    T.C. Hu, Multi-commodity network flows. Oper. Res. 11(3), 344–360 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  152. 166.
    R. Isaacs, Infinite families of non-trivial trivalent graphs which are not Tait colorable. Am. Math. Mon. 82, 221–239 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  153. 167.
    T. Jensen, G.F. Royle, Small graphs with chromatic number 5: a computer search. J. Graph Theory 19, 107–116 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  154. 168.
    D.A. Kappos, Strukturtheorie der Wahrscheinlichkeitsfelder und -Räume, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Heft 24 (Springer, Berlin, 1960)Google Scholar
  155. 169.
    H.A. Kierstead, J.H. Schmerl, The chromatic number of graphs which neither induce \(K_{1,3}\) nor \(K_5-e\). Discret. Math. 58, 253–262 (1986)zbMATHCrossRefGoogle Scholar
  156. 170.
    A.D. King, B.A. Reed, A. Vetta, An upper bound for the chromatic number of line graphs, Lecture given at EuroComb 2005, DMTCS Proc, AE, 151–156 (2005)Google Scholar
  157. 171.
    D. König, Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre. Math. Ann. 77, 453–465 (1916)MathSciNetzbMATHCrossRefGoogle Scholar
  158. 172.
    D. König, Graphen und Matrizen. Math. Fiz. Lapok 38, 116–119 (1931)Google Scholar
  159. 173.
    J. Körner, A property of conditional entropy. Studia Sci. Math. Hungar. 6, 355–359 (1971)MathSciNetzbMATHGoogle Scholar
  160. 174.
    J. Körner, An extension of the class of perfect graphs. Studia Sci. Math. Hungar. 8, 405–409 (1973)MathSciNetzbMATHGoogle Scholar
  161. 175.
    J.Körner, Coding of an information source having ambiguous alphabet and the entropy of graphs, in Transactions of the 6th Prague Conference on Information Theory, etc., 1971, Academia, Prague (1973), pp. 411–425Google Scholar
  162. 176.
    J. Körner, Fredman-Komlos bounds and information theory. SIAM J. Alg. Disc. Math. 7, 560–570 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  163. 177.
    J. Körner, G. Longo, Two-step encoding for finite sources. IEEE Trans. Inf. Theory 19, 778–782 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  164. 178.
    J. Körner, K. Marton, New bounds for perfect hashing via information theory. Eur. J. Comb. 9(6), 523–530 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  165. 179.
    J. Körner, K. Marton, Graphs that split entropies. SIAM J. Discret. Math. 1(1), 71–79 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  166. 180.
    J. Körner, A. Sgarro, A new approach to rate-distortion theory. Rend. Istit. Mat. Univ. di Trieste 18(2), 177–187 (1986)MathSciNetzbMATHGoogle Scholar
  167. 181.
    A.V. Kostochka, M. Stiebitz, Excess in colour-critical graphs, in Graph theory and combinatorial biology (Proceedings of Balatonlelle). Bolyai Society Mathematical Studies 7, 87–99 (1996)zbMATHGoogle Scholar
  168. 182.
    H.V. Kronk, The chromatic number of triangle-free graphs. Lecture Notes in Mathematics 303, 179–181 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  169. 183.
    H.W. Kuhn, Variants of the Hungarian method for assignment problems. Naval Res. Logist. Q. 3, 253–258 (1956)MathSciNetzbMATHCrossRefGoogle Scholar
  170. 184.
    E.L. Lawler, Optimal matroid intersections, in Combinatorial Structures and Their Applications, ed. by R. Guy, H. Hanani, N. Sauer, J. Schonheim (Gordon and Breach, 1970), p. 233Google Scholar
  171. 185.
    A. Lehman, On the width-length inequality, (Mimeo. 1965). Math. Program. 17, 403–413 (1979)zbMATHCrossRefGoogle Scholar
  172. 186.
    P.G.H. Lehot, An optimal algorithm to detect a line graph and output its root graph. J. Assoc. Comput. Mach. 21, 569–575 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  173. 187.
    C.G. Lekkerkerker, C.J. Boland, Representation of a finite graph by a set of intervals on the real line. Fund. Math. 51, 45–64 (1962)MathSciNetzbMATHGoogle Scholar
  174. 188.
    C. Linhares-Sales, F. Maffray, Even pairs in square-free Berge graphs, Laboratoire Leibniz Res. Rep. 51-2002 (2002)Google Scholar
  175. 189.
    N. Linial, Extending the Greene-Kleitman theorem to directed graphs. J. Comb. Theory Ser. A 30, 331–334 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  176. 190.
    L. Lovász, On chromatic number of finite set-systems. Acta Math. Acad. Sci. Hungar. 19, 59–67 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  177. 191.
    L. Lovász, Normal hypergraphs and the perfect graph conjecture. Discret. Math. 2(3), 253–267 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  178. 192.
    L. Lovász, A characterization of perfect graphs. J. Comb. Theory Ser. B 13, 95–98 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  179. 193.
    L. Lovász, On the Shannon capacity of a noisy channel. I.R.E. Trans. Inf. Theory 25, 1–7 (1979)Google Scholar
  180. 194.
    L. Lovász, Perfect graphs, in Selected Topics in Graph Theory, ed. by L.W. Beineke, R.J. Wilson, vol. 2, (Academic Press, New York, 1983), pp. 55–87Google Scholar
  181. 195.
    L. Lovász, Normal hypergraphs and the weak perfect graph conjecture, in Topics on Perfect Graphs, ed. by C. Berge, V. Chvátal. North-Holland Mathematics Studies, vol. 88 (North-Holland, Amsterdam, 1984), pp. 29–42 (Ann. Disc. Math. 21)Google Scholar
  182. 196.
    F. Maffray, B.A. Reed, A description of claw-free perfect graphs. J. Comb. Theory Ser. B 75(1), 134–156 (1999)Google Scholar
  183. 197.
    S.E. Markosjan, I.A. Karapetjan, Perfect graphs. Akad. Nauk Armjan. SSR Dokl. 63(5), 292–296 (1976)MathSciNetGoogle Scholar
  184. 198.
    K. Marton, On the Shannon capacity of probabilistic graphs. J. Comb. Theory Ser. B 57(2), 183–195 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  185. 199.
    R.J. McEliece, The Theory of Information and Coding, 2nd edn. Encyclopedia of Mathematics and its Applications, vol. 86 (Cambridge University Press, Cambridge, 2002)Google Scholar
  186. 200.
    R. Merris, Graph Theory (Wiley, New York, 2001)zbMATHGoogle Scholar
  187. 201.
    H. Meyniel, On the perfect graph conjecture. Discret. Math. 16(4), 339–342 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  188. 202.
    H. Meyniel, The graphs whose odd cycles have at least two chords, in Topics on Perfect Graphs, ed. by C. Berge, V. Chvátal (North-Holland, Amsterdam, 1984), pp. 115–120CrossRefGoogle Scholar
  189. 203.
    H. Meyniel, private communication with C. Berge, 1985 (or 1986?)Google Scholar
  190. 204.
    H. Meyniel, A new property of critical imperfect graphs and some consequences. Eur. J. Comb. 8, 313–316 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  191. 205.
    N.D. Nenov, On the small graphs with chromatic number 5 without 4-cliques. Discret. Math. 188, 297–298 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  192. 206.
    J. Nesetril, \(K\)-chromatic graphs without cycles of length \(\le 7\). Comment. Math. Univ. Carolina 7, 373–376 (1966)MathSciNetGoogle Scholar
  193. 207.
    S. Olariu, Paw-free graphs. Inf. Process. Lett. 28, 53–54 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  194. 208.
    O. Ore, Theory of Graphs. American Mathematical Society Colloquium publications, vol. 38 (American Mathematical Society, Providence, 1962)Google Scholar
  195. 209.
    M.W. Padberg, Perfect zero-one matrices. Math. Program. 6, 180–196 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  196. 210.
    K.R. Parthasarathy, G. Ravindra, The strong perfect graph conjecture is true for \(K_{1,3}\)-free graphs. J. Comb. Theory B 21, 212–223 (1976)zbMATHCrossRefGoogle Scholar
  197. 211.
    C. Payan, private communication with C. Berge (1981)Google Scholar
  198. 212.
    G. Polya, Aufgabe 424. Arch. Math. Phys. 20, 271 (1913)Google Scholar
  199. 213.
    M. Preissmann, C-minimal snarks. Ann. Discret. Math. 17, 559–565 (1983)zbMATHGoogle Scholar
  200. 214.
    H.J. Prömel, A. Steger, Almost all Berge graphs are perfect. Comb. Probab. Comput. 1(1), 53–79 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  201. 215.
    L. Rabern, On graph associations. SIAM J. Discret. Math. 20(2), 529–535 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  202. 216.
    L. Rabern, A note on Reed’s Conjecture, arXiv:math.CO/0604499 (2006)
  203. 217.
    J. Ramirez-Alfonsin, B. Reed (eds.), Perfect Graphs (Springer, Berlin, 2001)zbMATHGoogle Scholar
  204. 218.
    F.P. Ramsey, On a problem of formal logic. Proc. Lond. Math. Soc. 2(30), 264–286 (1930)Google Scholar
  205. 219.
    G. Ravindra, Strongly perfect line graphs and total graphs, in Finite and infinite sets, Vol. I, II, ed. by Eger, 1981; A. Hajnal, L. Lovász, V.T. Sós. Colloq. Math. Soc. János Bolyai, vol. 37, (North-Holland, Amsterdam, 1984), pp. 621–633Google Scholar
  206. 220.
    G. Ravindra, Research problems. Discret. Math. 80, 105–107 (1990)zbMATHCrossRefGoogle Scholar
  207. 221.
    G. Ravindra, D. Basavayya, Co-strongly perfect bipartite graphs. J. Math. Phys. Sci. 26, 321–327 (1992)MathSciNetzbMATHGoogle Scholar
  208. 222.
    G. Ravindra, D. Basavayya, Co-strongly perfect line graphs, in Combinatorial Mathematics and Applications (Calcutta, Sankhya Ser. A, vol. 54. Special Issue 1988, 375–381 (1988)Google Scholar
  209. 223.
    G. Ravindra, D. Basavayya, A characterization of nearly bipartite graphs with strongly perfect complements. J. Ramanujan Math. Soc. 9, 79–87 (1994)MathSciNetzbMATHGoogle Scholar
  210. 224.
    G. Ravindra, D. Basavayya, Strongly and costrongly perfect product graphs. J. Math. Phys. Sci. 29(2), 71–80 (1995)MathSciNetzbMATHGoogle Scholar
  211. 225.
    G. Ravindra, K.R. Parthasarathy, Perfect product graphs. Discret. Math. 20, 177–186 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  212. 226.
    B. Reed, \(\omega, \Delta \), and \(\chi \). J. Graph Theory 27(4), 177–212 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  213. 227.
    B. Reed, A strengthening of Brooks’ Theorem. J. Comb. Theory Ser. B 76(2), 136–149 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  214. 228.
    J.T. Robacker, Min-Max theorems on shortest chains and disjoint cuts of a network. Research Memorandum RM-1660, The RAND Corporation, Santa Monica, California (1956)Google Scholar
  215. 229.
    N. Robertson, P. Seymour, R. Thomas, Hadwiger’s conjecture for \(K_6\)-free graphs. Combinatorica 13, 279–361 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  216. 230.
    N. Robertson, P. Seymour, R. Thomas, Excluded minors in cubic graphs. manuscript (1996)Google Scholar
  217. 231.
    N. Robertson, P. Seymour, R. Thomas, Tutte’s edge-colouring conjecture. J. Comb. Theory Ser. B 70, 166–183 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  218. 232.
    N. Robertson, P. Seymour, R. Thomas, Permanents, Pfaffian orientations, and even directed circuits. Ann. Math. 150, 929–975 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  219. 233.
    F. Roussel, P. Rubio, About skew partitions in minimal imperfect graphs. J. Comb. Theory, Ser. B 83, 171–190 (2001)Google Scholar
  220. 234.
    N.D. Roussopoulos, A max \(\{m, n\}\) algorithm for determining the graph \(H\) from its line graph \(G\). Inf. Process. Lett. 2, 108–112 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  221. 235.
    B. Roy, Nombre chromatique et plus longs chemins. Rev. Fr. Automat. Inform. 1, 127–132 (1967)MathSciNetzbMATHGoogle Scholar
  222. 236.
    H. Sachs, On the Berge conjecture concerning perfect graphs, in Combinatorial Structures and their Applications (Proceedings of the Calgary International Conference, Calgary, Alberta) (Gordon and Breach, New York, 1969), pp. 377–384Google Scholar
  223. 237.
    M. Saks, A short proof of the the k-saturated partitions. Adv. Math. 33, 207–211 (1979)zbMATHCrossRefGoogle Scholar
  224. 238.
    J. Schönheim, Hereditary systems and Chvátal’s conjecture, in Proceedings of the Fifth British Combinatorial Conference (University of Aberdeen, Aberdeen, 1975), Congressus Numerantium, No. XV, Utilitas Math., Winnipeg, Man. (1976), pp. 537–539Google Scholar
  225. 239.
    D. Seinsche, On a property of the class of n-colorable graphs. J. Comb. Theory B 16, 191–193 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  226. 240.
    P. Seymour, Decomposition of regular matroids. J. Comb. Theory Ser. B 28, 305–359 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  227. 241.
    P. Seymour, Disjoint paths in graphs. Discret. Math. 29, 293–309 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  228. 242.
    P. Seymour, How the proof of the strong perfect graph conjecture was found. Gazette des Mathematiciens 109, 69–83 (2006)MathSciNetzbMATHGoogle Scholar
  229. 243.
    P. Seymour, K. Truemper, A Petersen on a pentagon. J. Comb. Theory Ser. B 72(1), 63–79 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  230. 244.
    S. Sridharan, On the Berge’s strong path-partition conjecture. Discret. Math. 112, 289–293 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  231. 245.
    L. Stacho, New upper bounds for the chromatic number of a graph. J. Graph Theory 36(2), 117–120 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  232. 246.
    M. Stehlík, Critical graphs with connected complements. J. Comb. Theory Ser. B 89(2), 189–194 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  233. 247.
    P. Stein, Chvátal’s conjecture and point intersections. Discret. Math. 43(2–3), 321–323 (1983)zbMATHCrossRefGoogle Scholar
  234. 248.
    P. Stein, J. Schönheim, On Chvátal’s conjecture related to hereditary systems. Ars Comb. 5, 275–291 (1978)zbMATHGoogle Scholar
  235. 249.
    F. Sterboul, Les parametres des hypergraphes et les problemes extremaux associes (Thèse, Paris, 1974), pp. 33–50Google Scholar
  236. 250.
    F. Sterboul, Sur une conjecture de V. Chvátal, in Hypergraph Seminar, ed. by C. Berge, D. Ray-Chaudhuri, Lecture Notes, in Mathematics, vol. 411, (Springer, Berlin, 1974), pp. 152–164Google Scholar
  237. 251.
    L. Surányi, The covering of graphs by cliques. Studia Sci. Math. Hungar. 3, 345–349 (1968)MathSciNetzbMATHGoogle Scholar
  238. 252.
    P.G. Tait, Note on a theorem in geometry of position. Trans. R. Soc. Edinb. 29, 657–660 (1880)zbMATHCrossRefGoogle Scholar
  239. 253.
    C. Thomassen, Five-coloring graphs on the torus. J. Comb. Theory B 62, 11–33 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  240. 254.
    K. Truemper, Alpha-balanced graphs and matrices and GF(3)-representability of matroids. J. Comb. Theory B 32, 112–139 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  241. 255.
    A. Tucker, Matrix characterizations of circular-arc graphs. Pac. J. Math. 39, 535–545 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  242. 256.
    A. Tucker, The strong perfect graph conjecture for planar graphs. Can. J. Math. 25, 103–114 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  243. 257.
    A. Tucker, Perfect graphs and an application to refuse collection. SIAM Rev. 15, 585–590 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  244. 258.
    A. Tucker, Structure theorems for some circular-arc graphs. Discret. Math. 7, 167–195 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  245. 259.
    A. Tucker, Coloring a family of circular arcs. SIAM J. Appl. Math. 29(3), 493–502 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  246. 260.
    A. Tucker, Critical perfect graphs and perfect \(3\)-chromatic graphs. J. Comb. Theory Ser. B 23(1), 143–149 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  247. 261.
    A. Tucker, The validity of the strong perfect graph conjecture for \(K_4\)-free graphs, in Topics on Perfect Graphs, ed. by C. Berge, V. Chvátal (1984), pp. 149–158 (Ann. Discret. Math. 21)Google Scholar
  248. 262.
    A. Tucker, Coloring perfect \((K_4-e)\)-free graphs. J. Comb. Theory Ser. B 42(3), 313–318 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  249. 263.
    W.T. Tutte, A short proof of the factor theorem for finite graphs. Can. J. Math. 6, 347–352 (1954)MathSciNetzbMATHCrossRefGoogle Scholar
  250. 264.
    W.T. Tutte, On the problem of decomposing a graph into \(n\) connected factors. J. Lond. Math. Soc. 36, 221–230 (1961)MathSciNetzbMATHCrossRefGoogle Scholar
  251. 265.
    W.T. Tutte, Lectures on matroids. J. Res. Nat. Bur. Stand. Sect. B 69B, 1–47 (1965)Google Scholar
  252. 266.
    W.T. Tutte, On the algebraic theory of graph colorings. J. Comb. Theory 1, 15–50 (1966)MathSciNetzbMATHCrossRefGoogle Scholar
  253. 267.
    P. Ungar, B. Descartes, Advanced problems and solutions: solutions: 4526. Am. Math. Mon. 61(5), 352–353 (1954)MathSciNetCrossRefGoogle Scholar
  254. 268.
    J. von Neumann, A certain zero-sum two-person game equivalent to the optimal assignment problem, in Contributions to the Theory of Games. Annals of Mathematics Studies, No. 28, vol. 2 (Princeton University Press, Princeton, 1953), pp. 5–12Google Scholar
  255. 269.
    K. Wagner, Über eine Eigenschaft der ebenen Komplexe. Math. Ann. 114, 570–590 (1937)MathSciNetzbMATHCrossRefGoogle Scholar
  256. 270.
    D.L. Wang, P. Wang, Some results about the Chvátal conjecture. Discrete Math. 24(1), 95–101 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  257. 271.
    D.B. West, Introduction to Graph Theory (Prentice-Hall, Englewood Cliffs, 1996)zbMATHGoogle Scholar
  258. 272.
    C. Witzgall, C.T. Zahn Jr., Modification of Edmonds’ maximum matching algorithm. J. Res. Nat. Bur. Stand. Sect. B 69B, 91–98 (1965)Google Scholar
  259. 273.
    Q. Xue, (\(C_4\), Lotus)-free Berge graphs are perfect. An. Stiint. Univ. Al. I. Cuza Iasi Inform. (N.S.) 4, 65–71 (1995)Google Scholar
  260. 274.
    Q. Xue, On a class of square-free graphs. Inf. Process. Lett. 57(1), 47–48 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  261. 275.
    A.A. Zykov, On some properties of linear complexes. Russian Math. Sbornik N. S. 24(66), 163–188 (1949)MathSciNetGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BielefeldBielefeldGermany

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