The Tutte Polynomial Part I: General Theory

  • Thomas Brylawski
Part of the C.I.M.E. Summer Schools book series (CIME, volume 83)


Matroid theory (sometimes viewed as the theory of combinatorial geometries or geometric lattices) is reasonably young as a mathematical theory (its traditional birthday is given as 1935 with the appearance of [159]) but has steadily developed over the years and shown accelerated growth recently due, in large part, to two applications. The first is in the field of algorithms. To coin an oversimplification: “when a good algorithm is known, a matroid structure is probably hidden away somewhere.” In any event, many of the standard good algorithms (such as the greedy algorithm) and many important ones whose complexities are currently being scrutinized (e.g., existence of a Hamiltonian path) can be thought of as matroid algorithms. In the accompanying lecture notes of Professor Welsh the connections between matroids and algorithms are presented.

Another important application of matroids is the theory of the Tutte polynomial
$$ {\text{t}}\left( {{\text{M;}}\,{\text{x, y}}} \right) = \sum {{\text{a}}_{{\text{ij}}} \left( {{\text{x - 1}}} \right)^{\text{i}} \left( {{\text{y - 1}}} \right)^{\text{j}} } $$


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arrowsmith, D. K. and Jaeger, F., “On the enumeration of chains in regular chain-groups ” (preprint, 1980).Google Scholar
  2. 2.
    Baclawski, K., “Whitney numbers of geometric lattices,” Advances in Math. 16 (1975), 125–138.MathSciNetMATHGoogle Scholar
  3. 3.
    ——, “The Möbius algebra as a Grothendieck ring,” J. of Algebra 57 (1979), 167–179.MathSciNetMATHGoogle Scholar
  4. 4.
    Barlotti, A., “Some topics in finite geometrical structures,” Institute of Statistics Mimeo Series No. 439, Department of Statistics, University of North Carolina, Chapel Hill, N. C, 1965.Google Scholar
  5. 5.
    ——, “Bounds for k-caps in PG(r,q) useful in the theory of error correcting codes,” Institute of Statistics Mimeo Series No. 484.2, Department of Statistics, University of North Carolina, Chapel Hill, N. C, 1966.Google Scholar
  6. 6.
    ——, “Results and problems in Galois geometry,” Colloquium on Combinatorics and its Applications, June, 1978, Colorado State University.Google Scholar
  7. 7.
    Bessinger, J. S., “On external activity and inversion in trees” (preprint).Google Scholar
  8. 8.
    Biggs, N., Algebraic Graph Theory, Cambridge University Press, 1974.Google Scholar
  9. 9.
    ——, “Resonance and reconstruction,” Proc. Seventh British Combinatorial Conference, Cambridge U. Press, 1979, 1–21.Google Scholar
  10. 10.
    Birkhoff, G. D., “A Determinant formula for the number of ways of coloring a map,” Ann. of Math. (2) 14 (1913), 42–46.MathSciNetGoogle Scholar
  11. 11.
    Birkhoff, G. D. and Lewis, D. C., “Chromatic polynomials,” Trans. Amer. Math. Soc. 60 (1946), 355–451.MathSciNetMATHGoogle Scholar
  12. 12.
    Bixby, R. E., “A omposition for matroids,” J. Comb. Th. (B) 18 (1975), 59–73.MathSciNetMATHGoogle Scholar
  13. 13.
    Björmer, A., “On the homology of geometric lattices,” (preprint: 1977 No. 9, Matematiska Institutionen Stockholms Universitet, Stockholm, Sweden).Google Scholar
  14. 14.
    ——, “Homology of matroids ” (preprint, to appear Combinatorial Geometries, H. Crapo, G.-C. Rota, N. White eds.).Google Scholar
  15. 15.
    Björner, A., “Some matroid inequalities,” Disc. Math. 31 (1980), 101–103.MATHGoogle Scholar
  16. 16.
    Bland, R. G. and Las Vergnas, M., “Orientability of matroids,” J. Comb. Th. (B) 24 (1978), 94–123.MathSciNetMATHGoogle Scholar
  17. 17.
    Bondy, J. A. and Hemminger, R. L., “Graph reconstruction A survey,” Research Report CORR 76–49, Dept. of Comb. and Opt., University of Waterloo, Waterloo, Ontario, Canada, 1976.Google Scholar
  18. 18.
    Bondy, J. A. and Murty, U. S. R., Graph Theory with Applications, Macmillan, London; American Elsevier, New York, 1976.Google Scholar
  19. 19.
    Brini, A., “A class of rank-invariants for perfect matroid designs,” Europ. J. Comb. 1 (1980), 33–38.MathSciNetMATHGoogle Scholar
  20. 20.
    Brooks, R. L., “On colouring the nodes of a network,” Proc. Cambridge Phil. Soc. 37 (1941), 194–197.MathSciNetGoogle Scholar
  21. 21.
    Brouwer, A. E. and Schriver, A., “The blocking number of an affine space,” J. Comb. Th. (A) 24 (1978), 251–253.MATHGoogle Scholar
  22. 22.
    Bruen, A. A. and de Resmini, M., “Blocking sets in affine planes” (preprint, 1981).Google Scholar
  23. 23.
    Bruen, A. A. and Thas, J. A., “Blocking sets,” Geom. Dedic. 6 (1977), 193–203.MathSciNetMATHGoogle Scholar
  24. 24.
    Brylawski, T., “A Combinatorial model for series-parallel networks,” Transactions of the AMS, 154 (1971), 1–22.MathSciNetMATHGoogle Scholar
  25. 25.
    ——, “Some properties of basic families of subsets,” Disc. Math. 6 (1973), 333–341.MathSciNetMATHGoogle Scholar
  26. 26.
    ——, “The Tutte-Grothendieck ring,” Algebra Universalis 2 (1972), 375–388.MathSciNetMATHGoogle Scholar
  27. 27.
    ——, “A Decomposition for combinatorial geometries,” Transactions of the AMS, 171 (1972), 235–282.MathSciNetMATHGoogle Scholar
  28. 28.
    ——, “Reconstructing combinatorial geoemetries,” Graphs and Combinatorics, Springer-Verlag, Lecture Notes in Mathematics 406 (1974), 226–235.MathSciNetGoogle Scholar
  29. 29.
    ——, “Modular constructions for combinatorial geometries,” Transactions of AMS, 203 (1975), 1–44.MathSciNetMATHGoogle Scholar
  30. 30.
    ——, “On the nonreconstructibility of combinatorial geometries,” Journal of Comb. Theory (B), 19 (1975), 72–76.MathSciNetMATHGoogle Scholar
  31. 31.
    Brylawski, T., “An Affine representation for transversal geometries,” Studies in Applied Mathematics, 54 (1975), 143–160.MathSciNetMATHGoogle Scholar
  32. 32.
    ——, “A Combinatorial perspective on the Radon convexity theorem”, Geometriae Dedicata, 5 (1976), 459–466.MathSciNetMATHGoogle Scholar
  33. 33.
    ——, “A Determinantal identity for resistive networks,” SIAM J. Appl. Math., 32 (1977), 443–449.MathSciNetMATHGoogle Scholar
  34. 34.
    ——, “Connected matroids with smallest Whitney numbers,” Discrete Math. 18 (1977), 243–252.MathSciNetMATHGoogle Scholar
  35. 35.
    ——, ”The Broken-circuit complex,” Transactions of AMS, 234 (1977), 417–433.MathSciNetMATHGoogle Scholar
  36. 36.
    ——, “Geometrie combinatorie e Loro applicazioni” (1977). “Funzioni di Möbius” (1977). “Teoria dei Codici e matroidi” (1979). “Matroidi coordinabili” (1981). University of Rome Lecture Series.Google Scholar
  37. 37.
    ——, “Intersection theory for embeddings of matroids into uniform geometries,” Studies in Applied Mathematics 61 (1979), 211”244.MathSciNetMATHGoogle Scholar
  38. 38.
    ——, “The Affine dimension of the space of intersection matrices,” Rendiconti di Mathematics 13 (1980), 59–68.MathSciNetMATHGoogle Scholar
  39. 39.
    ——, “Intersection theory for graphs,” J. Comb. Th. (B) 30 (1981), 233–246.MathSciNetMATHGoogle Scholar
  40. 40.
    ——, “Hyperplane reconstruction of the Tutte polynomial of a geometric lattice,” Discrete Math. 35 (1981), 25–38.MathSciNetMATHGoogle Scholar
  41. 41.
    Brylawski, T. and Kelly, D., “Matroids and combinatorial geometries,” Studies in Combinatorics, G.-C. Rota, ed., Math. Association of America, 1978.Google Scholar
  42. 42.
    x2014;—, Matroids and Combinatorial Geometries, Carolina Lecture Series Volumn 8, Chapel Hill, N. C., 1980.MATHGoogle Scholar
  43. 43.
    Brylawski, T., Lo Re, P. M., Mazzocca, F., and Olanda, D., “Alcune applicazioni della Teoria dell' intersezione alle geometrie di Galois,” Ricerche di Matematica 29 (1980), 65–84.MathSciNetMATHGoogle Scholar
  44. 44.
    Brylawski, T. and Lucas, T. D., “Uniquely representable combinatorial geometries,” Proceedings of the Colloquio Internazionale sul tema Teorie Combinatorie, Rome, 1973, Atti Dei Convegni Lincei 17, Tomo I (1976), 83–104.Google Scholar
  45. 45.
    Brylawski, T. and Oxley, J., “The Broken-circuit complex: its structure and factorizations,” European J. Combinatorics 2 (1981), 107–121.MathSciNetMATHGoogle Scholar
  46. 46.
    ——, “Several identities for the characteristic polynomial of a combinatorial geometry,” Discrete Math. 31 (1980), 161–170.MathSciNetMATHGoogle Scholar
  47. 47.
    Cardy, S., “The Proof of and generalisations to a conjecture by Baker and Essam,” Discrete Math. 4 (1973), 101–122.MathSciNetMATHGoogle Scholar
  48. 48.
    Cordovil, R., “Contributions à la théorie des géométries combinatories,” Thesis, 1'Université Pierre et Marie Curie, Paris, France.Google Scholar
  49. 49.
    ——, “Sur 1'evaluation t(M;2,0) du polynome de Tutte d'un matroïde et une conjecture de B. Griinbaum relative aux arrangements de droites du plan ” (preprint, 1980).Google Scholar
  50. 50.
    Cordovil, R., Las Vergnas, M., and Mandel, A., “Euler's relation, Mobius functions, and matroid identities” (preprint, 1980).Google Scholar
  51. 51.
    Cossu, A., “Su alcune propretà dei {k,n}-archi di un piano proiettivo sopra un corpo finito,” Rend. di Mat. (5), 20 (1961), 271–277.MathSciNetMATHGoogle Scholar
  52. 52.
    Crapo, H. H., “The Mobius function of a lattice,” J. Comb. Th. 1 (1966), 126–131.MathSciNetMATHGoogle Scholar
  53. 53.
    x2014;—, “A Higher invariant for matroids,” J. Comb. Th. 2 (1967), 406–417.MathSciNetMATHGoogle Scholar
  54. 54.
    x2014;—, “Möbius inversion in lattices,” Archiv. der Math. 19 (1968), 595–607.MathSciNetGoogle Scholar
  55. 55.
    ——, “The Joining of exchange geometries,” J. Math. Mech. 17 (1968), 837–852.MathSciNetMATHGoogle Scholar
  56. 56.
    ——, “The Tutte polynomial,” Aequationes Math. 3 (1969), 211–229.MathSciNetMATHGoogle Scholar
  57. 57.
    ——, “Chromatic polynomials for a join of graphs,” Colloquia Mathematica Societatis János Bolyai, Combinatorial Theory and its Applications, Balatonfüred (Hungary), 1969, 239–245.Google Scholar
  58. 58.
    x2014;—, “Erecting geometries,” Proceedings of 2nd Chapel Hill Conference on Combinatorial Math. (1970), 74–99.Google Scholar
  59. 59.
    ——, “Constructions in combinatorial geometries,” (N.S.F. Advanced Science Seminar in Combinatorial Theory) (Notes, Bowdoin College), 1971).Google Scholar
  60. 60.
    Crapo, H. H. and Rota, G.-C, “On the Foundations of Combinatorial Theory: Combinatorial Geometries (preliminary edition), M.I.T. Press, 1970.MATHGoogle Scholar
  61. 61.
    d'Antona, 0. and Kung, J. P. S., “Coherent orientations and series-parallel networks,” Disc. Math. 32 (1980), 95–98.MathSciNetMATHGoogle Scholar
  62. 62.
    Deza, M., “On perfect matroid designs,” Proc. Kyoto Conference, 1977, 98–108.Google Scholar
  63. 63.
    Deza, M. and Singi, N. M., “Some properties of perfect matroid designs,” Ann. Disc. Math. 6 (1980).Google Scholar
  64. 64.
    Dirac, G. A., “A roperty of 4-chromatic graphs and some remarks on critical graphs,” J. London Math. Soc. 27 (1952), 85–92.MathSciNetMATHGoogle Scholar
  65. 65.
    Dowling, T. A., “Codes, packings and the critical problem,” Atti del Convegno di Geometria Combinatoria e sue Applicazioni (Perugia, 1971), 210–224.Google Scholar
  66. 66.
    ——, “A Class of geometric lattices based on finite groups,” J. Comb. Th. 13, (1973), 61–87.MathSciNetGoogle Scholar
  67. 67.
    x2014;—, “A q-analog of the partition lattice,” A Survey of Combinatorial Theory, North Holland (1973), 101–115.Google Scholar
  68. 68.
    Dowling, T. A. and Wilson, R. M., “The Slimmest geometric lattices,” Trans. Amer. Math. Soc. 196 (1974), 203–215.MathSciNetMATHGoogle Scholar
  69. 69.
    Edmonds, J. and Fulkerson, D. R., “Transversals and matroid partition,” J. Res. Nat. Bur. Stand. 69B (1965), 147–153.MathSciNetGoogle Scholar
  70. 70.
    Edmonds, J., Murty, U. S. R., and Young, P., “Equicardinal matroids and matroid designs,” Combinatorial Mathematics and its Applications, Chapel Hill, N. C., (1970), 498–582.Google Scholar
  71. 71.
    Essam, J. W., “Graph theory and statistical physics,” Discrete Math. 1 (1971), 83–112.MathSciNetMATHGoogle Scholar
  72. 72.
    Goldman, J. and Rota, G.-C, “The Number of subspaces of a vector space,” Recent Progress in Combinatorics, Academic Press, New York, 1969, 75–83.Google Scholar
  73. 73.
    Greene, C., “An Inequality for the Möbius function of a geometric lattice,” Proc. Conf. on Möbius Algebras (Waterloo), 1971; also: Studies in Appl. Math. 54 (1975), 71–74.MathSciNetMATHGoogle Scholar
  74. 74.
    x2014;—, “On the Mobius algebra of a partially ordered set,” Advances in Math. 10 (1973), 177–187.MathSciNetMATHGoogle Scholar
  75. 75.
    ——, “Weight enumeration and the geometry of linear codes,” Studies in Appl. Math. 55 (1976), 119–128.MathSciNetMATHGoogle Scholar
  76. 76.
    ——, “Acyclic orientations,” (Notes), Higher Combinatorics, M. Aigner, ed., D. Reidel, Dordrecht (1977), 65–68.Google Scholar
  77. 77.
    Greene, C. and Zaslavsky, T., “On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and acyclic orientations of graphs ” (preprint, 1980).Google Scholar
  78. 78.
    Greenwell, D. L. and Hemminger, R. L., “Reconstructing graphs,” The Many Facets of Graph Theory, Springer-Verlag, Berlin, 1969, 91–114.Google Scholar
  79. 79.
    Hardy, G. H., Littlewood, J. E., and Pólya, G., Inequalities, Cambridge U. Press, 1934.Google Scholar
  80. 80.
    Heron, A. P., “Matroid polynomials,” Combinatorics (Institute of Math. & Appl.) D. J. A. Welsh and D. R. Woodall, eds., 164–203.Google Scholar
  81. 81.
    Hsieh, W. N. and Kleitman, D. J., “Normalized matching in direct products of partial orders,” Studies in Applied Math. 52 (1973), 285–289.MathSciNetMATHGoogle Scholar
  82. 82.
    ——, “Flows and generalized coloring theorems in graphs,” J. Comb. Th. (B) 26 (1979), 205–216.Google Scholar
  83. 83.
    ——, “A Constructive approach to the critical problem ” (to appear: Europ. J_. Combinatorics, 1981).Google Scholar
  84. 84.
    Kahn, J. and Kung, J. P. S., “Varieties and universal models in the theory of combinatorial geometries,” Bulletin of the AMS 3 (1980), 857–858.MathSciNetMATHGoogle Scholar
  85. 85.
    Kelly, D. G. and Rota, G.-C, “Some problems in combinatorial geometry,” A. Survey of Combinatorial Theory, North Holland, 1973, 309–313.Google Scholar
  86. 86.
    Knuth, D. E., “The Asymptotic number of geometries,” J. Comb. Th. (A) 17 (1974), 398–401.Google Scholar
  87. 87.
    Las Vergnas, M., “Matroids orientables,” C. R. Acad. Sci. (Paris), 280A (1975), 61–64.MathSciNetGoogle Scholar
  88. 88.
    ——, “Extensions normales d'un matroide, polynôme de Tutte d'un morphisme,” C. R. Acad. Sci. (Paris), 280 (1975), 1479–1482.MathSciNetMATHGoogle Scholar
  89. 89.
    ——, “Acyclic and totally cyclic orientations of combinatorial geometries,” Disc. Math., 20 (1977), 51–61.MathSciNetGoogle Scholar
  90. 90.
    ——, “Sur les activités des orientations d'une geometrie combinatoire,” Collogue Mathématiques Discrètes: Codes et Hypergraphes, Bruxelles, 1978, 293–300.Google Scholar
  91. 91.
    ——, “Eulerian circuits of 4-valent graphs imbedded in surfaces,” Colloquia Mathematica Societatis János Bolyai 25, Algebraic Methods in Graph Theory, Szeged (Hungary), 1978, 451–477.Google Scholar
  92. 92.
    Las Vergnas, M., “On Eulerian partitions of graphs,” Graph Theory and Combinatorics, R. J. Wilson (ed.), Research Notes in Math. 34, Pitman Advanced Publishing Program, 1979.Google Scholar
  93. 93.
    ——, “On the Tutte polynomial of a morphism of matroids,” Proc. Joint Canada-France Combinatorial Colloquium, Montréal 1979, Annals Discrete Math. 8 (1980), 7–20.MathSciNetMATHGoogle Scholar
  94. 94.
    Lindner, C. C. and Rosa, A., “Steiner quadruple systems a survey,” Discrete Math. 22:147–181 (1978).MathSciNetMATHGoogle Scholar
  95. 95.
    Lindström, B., “On the chromatic number of regular matroids,” J. Comb. Theory (B) 24 (1978), 367–369.MATHGoogle Scholar
  96. 96.
    Lucas, T. D., “Properties of rank preserving weak maps,” A.M.S. Bull. 80 (1974), 127–131.MathSciNetMATHGoogle Scholar
  97. 97.
    ——, “Weak maps of combinatorial geometries,” Trans. Am. Math. Soc. 206 (1975), 247–279.MathSciNetMATHGoogle Scholar
  98. 98.
    Macwilliaras, F. J., “A Theorem on the distribution of weights in a systematic code,” Bell System Tech. J. 42 (1963), 79–94.MathSciNetGoogle Scholar
  99. 99.
    Martin, P., “Enumerations eulériennes dans les multigraphes et invariants de Tutte-Grothendieck,” Thesis, Grenoble, 1977.Google Scholar
  100. 100.
    ——, “Remarkable valuation of the dichromatic polynomial of planar multigraphs,” J. Comb. Th. (B) 24 (1978), 318–324.MATHGoogle Scholar
  101. 101.
    Mason, J., “Matroids: unimodal conjectures and Motzkin's theorem,” Combinatorics (Institute of Math. & Appl.) (D. J. A. Welsh and D. R. Woodall, eds., 1972), 207–221.Google Scholar
  102. 102.
    ——, “Matroids as the study of geometrical configurations,” Higher Combinatorics, M. Aigner, ed., D. Reidel, Dordrecht, Holland, 1977, 133–176.Google Scholar
  103. 103.
    Minty, G. J., “On the axiomatic foundations of the theories of directed linear graphs, electrical networks and network programming,” Journ. Math. Mech. 15 (1966), 485–520.MathSciNetMATHGoogle Scholar
  104. 104.
    Mullin, R. C. and Stanton, R. G., “A Covering problem in binary spaces of finite dimension,” Graph Theory and Related Topics (J. A. Bondy and U.S.R. Murty, eds.) Academic Press, New York, 1979.Google Scholar
  105. 105.
    Hurty, U.S.R., “Equicardinal matroids,” J. Comb. Th. 11 (1971), 120–126.MATHGoogle Scholar
  106. 106.
    Nash-Williams, C. St. J.A., “An Application of matroids to graph theory,” Theory of Graphs International Symposium (Rome), Dunod (Paris) (1966), 263–265.Google Scholar
  107. 107.
    Oxley, J. G., “Colouring, packing and the critical problem,” Quart. J. Math. Oxford, (2), 29, 11–22.Google Scholar
  108. 108.
    ——, “Cocircuit coverings and packings for binary matroids,” Math. Proc. Cambridge Philos. Soc. 83 (1978), 347–351.MathSciNetMATHGoogle Scholar
  109. 109.
    ——, “On cographic regular matroids,” Discrete Math. 25 (1979), 89–90.MathSciNetMATHGoogle Scholar
  110. 110.
    ——, “A Generalization of a covering problem of Mullin and Stanton for matroids,” Combinatorial Mathematics VI. Edited by A. F. Horadam and W. D. Wallis, Lecture Notes in Mathematics Vol. 748, Springer-Verlag, Berlin, Heidelberg, New York, 1979, 92–97.Google Scholar
  111. 111.
    ——, “On a covering problem of Mullin and Stanton for binary matroids,” Aequationes Math. 19 (1979), 118, and 20 (1980), 104–112.MathSciNetGoogle Scholar
  112. 112.
    ——, “On Crapo's beta invariant for matroids,” Studies in Appl. Math. (to appear).Google Scholar
  113. 113.
    ——, “On a matroid identity” (preprint, 1981).Google Scholar
  114. 114.
    Oxley, J. G., Prendergast, K. and Row, D. H., “Matroids whose ground sets are domains of functions ” (to aopear, J Austral Math. Soc. (A).)Google Scholar
  115. 115.
    Oxley, J. G. and Welsh, D. J. A., “On some percolation results of J. M. Hammersley,” J. Appl. Probability 16 (1979), 526–540.MathSciNetMATHGoogle Scholar
  116. 116.
    ——, and ——, “The Tutte polynomial and percolation,” Graph Theory and Related Topics. Edited by J. A. Bondy and U.S.R. Murty, Academic Press, New York, San Francisco, London, 1979, 329–339.Google Scholar
  117. 117.
    Read, R. C, “An Introduction to chromatic polynomials,” J. Comb. Th., 4 (1968), 52–71.MathSciNetGoogle Scholar
  118. 118.
    Rota, G.-C, “On the foundations of combinatorial theory I,” Z. Wahrsch, 2 (1964), 340–368.MathSciNetMATHGoogle Scholar
  119. 119.
    ——, “Combinatorial analysis as a theory,” Hedrick Lectures, Math. Assoc, of Amer., Summer Meeting, Toronto, 1967.Google Scholar
  120. 120.
    ——, “Combinatorial theory, old and new,“ Int. Cong. Math. (Nice) (1970) 3, 229–233.Google Scholar
  121. 121.
    Scafati Tallini, M., “{k,n}-archi di un piano grafico finito, con particolare riguardo a quelli con due caratteri, Nota I, II,” Rend. Acc. Naz. Lincei 40 (8) (1966), 812–818, 1020–1025.Google Scholar
  122. 122.
    ——, “Calotte di tipo (m,n) in uno spazio di Galois sr,q,” Rend. Acc. Naz. Lincei 53(8) (1973), 71–81.MATHGoogle Scholar
  123. 123.
    Segre, B., Lectures on Modern Geometry, Edizioni Creomonese, Roma, 1961.MATHGoogle Scholar
  124. 124.
    Seymour, P. D., “On Tutte's extension of the four-colour problem ” (preprint, 1979).Google Scholar
  125. 125.
    ——, “Decomposition of regular matroids,” J. Comb. Th. (B) 28 (1980), 305–359.Google Scholar
  126. 126.
    ——, “Nowhere-zero 6-flows,” J. Comb. Th. (B) 30 (1981), 130–135.Google Scholar
  127. 127.
    Seymour, P. D. and Welsh, D. J. A., “Combinatorial applications of an inequality from statistical mechanics,” Math. Proc. Cambridge Phil. Soc. 77 (1975), 485–497.MathSciNetMATHGoogle Scholar
  128. 128.
    Shepherd, G. C., “Combinatorial properties of associated zonotopes,” Can. J. Math. 26 (1974), 302–321.Google Scholar
  129. 129.
    Smith, C. A. B., “Electric currents in regular matroids,” Combinatorics (Institute of Math. & Appl.) (D. J. A. Welsh & D. R. Woodall, eds., 1972), 262–285.Google Scholar
  130. 130.
    ——, “Patroids,” J. Comb. Th. 16 (1974), 64–76.Google Scholar
  131. 131.
    Stanley, R., “Modular elements of geometric lattices,” Algebra Universalis, 1 (1971), 214–217.MathSciNetMATHGoogle Scholar
  132. 132.
    ——, “Supersolvable semimodular lattices,” Proc. Conference on Möbius Algebras, University of Waterloo, 1971, pp. 80–142.Google Scholar
  133. 133.
    ——, “Supersolvable lattices,” Alg. Universalis 2 (1972), 197–217.MathSciNetMATHGoogle Scholar
  134. 134.
    ——, “Acyclic orientations of graphs,” Disc. Math. 5 (1974), 171–178.MathSciNetGoogle Scholar
  135. 135.
    Szekeres, G. and Wilf, H., “An Inequality for the chromatic number of a graph,” J. Comb. Th. 4 (1968), 1–3.MathSciNetGoogle Scholar
  136. 136.
    Tallini, G., “Problemi e risultati sulle geometrie di Galois,” Rel. N. 30, 1st. di Mat. dell' Univ. di Napoli (1973).Google Scholar
  137. 137.
    Tutte, W. T., “A Ring in graph theory,” Proc. Cambridge Phil Soc. 43 (1947), 26–40.MathSciNetMATHGoogle Scholar
  138. 138.
    ——, “A Contribution to the theory of chromatic polynomials,” Canad. J. Math. 6 (1954), 80–91.MathSciNetMATHGoogle Scholar
  139. 139.
    ——, “A Class of Abelian groups,” Canad. J. Math. 8 (1956), 13–28.MathSciNetMATHGoogle Scholar
  140. 140.
    ——, “Matroids and graphs,” Trans. Amer. Math. Soc. 90 (1959), 527–552.MathSciNetMATHGoogle Scholar
  141. 141.
    ——, “Lectures on matroids,” J. Res. Nat. Bur. Stand. 69B (1965), 1–48.MathSciNetGoogle Scholar
  142. 142.
    ——, “On the algebraic theory of graph coloring,” J. Comb. Th. 1 (1966), 15–50.MathSciNetMATHGoogle Scholar
  143. 143.
    ——, “On dichromatic polynomials,” J. Comb. Th. 2(1967), 301–320.MathSciNetMATHGoogle Scholar
  144. 144.
    ——, “Projective geometry and the 4-color problem,” Recent Progress in Combinatorics (W. T. Tutte, ed.) Academic Press 1969, 199–207.Google Scholar
  145. 145.
    ——, “Codichromatic graphs,” J. Comb. Th. 16 (1974), 168–175.MathSciNetMATHGoogle Scholar
  146. 146.
    ——, “All the king's men (a guide to reconstruction),” Graph Theory and Related Topics, Academic Press, 1979, 15–33.Google Scholar
  147. 147.
    Van Lint, J. H., Coding Theory, Springer Lecture Notes, 201, (1971).Google Scholar
  148. 148.
    Walton, P. N. and Welsh, D. J. A., “On the chromatic number of binary matroids,” Mathematika 27 (1980), 1–9.MathSciNetMATHGoogle Scholar
  149. 149.
    Welsh, D. J. A., “Euler and bipartite matroids,” J. Comb. Th. 6 (1969), 375–377.MathSciNetMATHGoogle Scholar
  150. 150.
    ——, “Combinatorial problems in matroid theory,” Combinatorial Mathematics and its Applications, Academic Press, (1971), 291–307.Google Scholar
  151. 151.
    ——, Matroid Theory, Academic Press, London, 1976.MATHGoogle Scholar
  152. 152.
    ——, “Percolation and related topoics,” Science Progress 64 (1977).Google Scholar
  153. 153.
    ——, “Colouring problems and matroids,“ Proc. Seventh British Combinatorial Conference, Cambridge U. Press (1979), 229–257.Google Scholar
  154. 154.
    Welsh, D. J. A., “Colourings, flows and projective geometry,” Nieuw Archief voor Wiskunde (3), 28 (1980), 159–176.MathSciNetMATHGoogle Scholar
  155. 155.
    White, N., “The Critical problem and coding theory,” Research Paper, SPS-66 Vol. III, Section 331, Jet Propulsion Laboratory, Pasadena, CA. (1972).Google Scholar
  156. 156.
    Whitney, H., “A Logical expansion in mathematics,” Bull. Amer. Math. Soc. 38 (1932), 572–579.MathSciNetGoogle Scholar
  157. 157.
    ——, “The Coloring of graphs,” Annals of Math. 33 (1932), 688–718.MathSciNetGoogle Scholar
  158. 158.
    ——, “2-isomorphic graphs,” Amer. J. Math. 55 (1933), 245–254.MathSciNetGoogle Scholar
  159. 159.
    ——, “On the abstract properties of linear dependence,” Amer. J. Math. 57 (1935), 509–533.MathSciNetGoogle Scholar
  160. 160.
    Wilf, H. S., “Which polynomials are chromatic?” Atti dei Convegni Lincei 17, Tomo 1 (1976), 247–256.Google Scholar
  161. 161.
    Winder, R. O., “Partitions of n-space by hyperplanes,” SlAM J. Appl. Math. 14 (1966), 811–818.MathSciNetMATHGoogle Scholar
  162. 162.
    Young, P. and Edmonds, J., “Matroid designs,” J. Res. Nat. Bur. Stan. 72B (1972), 15–44.Google Scholar
  163. 163.
    Zaslavsky, T., “Facing up to arrangements: face count formulas for partitions of space by hyperplanes,” Memoirs Amer. Math. Soc. 154 (1975).Google Scholar
  164. 164.
    ——, “Counting faces of cut-up spaces,“ Bull. Amer. Math. Soc. 81 (1975), 916–918.MathSciNetMATHGoogle Scholar
  165. 165.
    ——, “Maximal dissections of a simplex,” J. Comb. Th. (A) 20 (1976), 244–257.MathSciNetMATHGoogle Scholar
  166. 166.
    ——, “The Möbius function and the characteristic polynomial” (preprint: chapter for Combinatorial Geometries, H. Crapo, G.-C. Rota, and N. White eds.).Google Scholar
  167. 167.
    ——, “Arrangements of hyperplanes; matroids and graphs,” Proc. Tenth S.E. Conf. on Combinatorics, Graph Theory and Computing (Boca Raton, 1979), Vol. II, 895–911, Utilitas Math. Publ. Co., Winnipeg, Man., 1979.MathSciNetGoogle Scholar
  168. 168.
    ——, “The Geometry of root systems and signed graphs,” Amer. Math. Monthly, 88 (1981), 88–105.MathSciNetMATHGoogle Scholar
  169. 169.
    Zaslavsky, T., “Signed graphs” (preprint, 1980).Google Scholar
  170. 170.
    ——, “Orientation of signed graphs” (preprint, 1980).Google Scholar
  171. 171.
    ——, “Signed graph coloring” (preprint, 1980).Google Scholar
  172. 172.
    ——, “Chromatic invariants of signed graphs” (preprint, 1980).Google Scholar
  173. 173.
    ——, “Bicircular geometry and the lattice of forest of a graph” (preprint, 1980).Google Scholar
  174. 174.
    ——, “The slimmest arrangements of hyperplanes: I. Geometric lattices and projective arrangements” (preprint, 1980).Google Scholar
  175. 175.
    ——, “The slimmest arrangements of hyperplanes: II. Basepointed geometric lattices and Euclidean arrangements (preprint, 1980).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Thomas Brylawski
    • 1
  1. 1.Department of MathematicsUniversity of North CarolinaUSA

Personalised recommendations