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The Tutte Polynomial Part I: General Theory

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Matroid Theory and its Applications

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

Abstract

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

$$ {\text{t}}\left( {{\text{M;}}\,{\text{x, y}}} \right) = \sum {{\text{a}}_{{\text{ij}}} \left( {{\text{x - 1}}} \right)^{\text{i}} \left( {{\text{y - 1}}} \right)^{\text{j}} } $$

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Bibliography

  1. Arrowsmith, D. K. and Jaeger, F., “On the enumeration of chains in regular chain-groups ” (preprint, 1980).

    Google Scholar 

  2. Baclawski, K., “Whitney numbers of geometric lattices,” Advances in Math. 16 (1975), 125–138.

    MathSciNet  MATH  Google Scholar 

  3. ——, “The Möbius algebra as a Grothendieck ring,” J. of Algebra 57 (1979), 167–179.

    MathSciNet  MATH  Google Scholar 

  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. ——, “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. ——, “Results and problems in Galois geometry,” Colloquium on Combinatorics and its Applications, June, 1978, Colorado State University.

    Google Scholar 

  7. Bessinger, J. S., “On external activity and inversion in trees” (preprint).

    Google Scholar 

  8. Biggs, N., Algebraic Graph Theory, Cambridge University Press, 1974.

    Google Scholar 

  9. ——, “Resonance and reconstruction,” Proc. Seventh British Combinatorial Conference, Cambridge U. Press, 1979, 1–21.

    Google Scholar 

  10. Birkhoff, G. D., “A Determinant formula for the number of ways of coloring a map,” Ann. of Math. (2) 14 (1913), 42–46.

    MathSciNet  Google Scholar 

  11. Birkhoff, G. D. and Lewis, D. C., “Chromatic polynomials,” Trans. Amer. Math. Soc. 60 (1946), 355–451.

    MathSciNet  MATH  Google Scholar 

  12. Bixby, R. E., “A omposition for matroids,” J. Comb. Th. (B) 18 (1975), 59–73.

    MathSciNet  MATH  Google Scholar 

  13. Björmer, A., “On the homology of geometric lattices,” (preprint: 1977 No. 9, Matematiska Institutionen Stockholms Universitet, Stockholm, Sweden).

    Google Scholar 

  14. ——, “Homology of matroids ” (preprint, to appear Combinatorial Geometries, H. Crapo, G.-C. Rota, N. White eds.).

    Google Scholar 

  15. Björner, A., “Some matroid inequalities,” Disc. Math. 31 (1980), 101–103.

    MATH  Google Scholar 

  16. Bland, R. G. and Las Vergnas, M., “Orientability of matroids,” J. Comb. Th. (B) 24 (1978), 94–123.

    MathSciNet  MATH  Google Scholar 

  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. Bondy, J. A. and Murty, U. S. R., Graph Theory with Applications, Macmillan, London; American Elsevier, New York, 1976.

    Google Scholar 

  19. Brini, A., “A class of rank-invariants for perfect matroid designs,” Europ. J. Comb. 1 (1980), 33–38.

    MathSciNet  MATH  Google Scholar 

  20. Brooks, R. L., “On colouring the nodes of a network,” Proc. Cambridge Phil. Soc. 37 (1941), 194–197.

    MathSciNet  Google Scholar 

  21. Brouwer, A. E. and Schriver, A., “The blocking number of an affine space,” J. Comb. Th. (A) 24 (1978), 251–253.

    MATH  Google Scholar 

  22. Bruen, A. A. and de Resmini, M., “Blocking sets in affine planes” (preprint, 1981).

    Google Scholar 

  23. Bruen, A. A. and Thas, J. A., “Blocking sets,” Geom. Dedic. 6 (1977), 193–203.

    MathSciNet  MATH  Google Scholar 

  24. Brylawski, T., “A Combinatorial model for series-parallel networks,” Transactions of the AMS, 154 (1971), 1–22.

    MathSciNet  MATH  Google Scholar 

  25. ——, “Some properties of basic families of subsets,” Disc. Math. 6 (1973), 333–341.

    MathSciNet  MATH  Google Scholar 

  26. ——, “The Tutte-Grothendieck ring,” Algebra Universalis 2 (1972), 375–388.

    MathSciNet  MATH  Google Scholar 

  27. ——, “A Decomposition for combinatorial geometries,” Transactions of the AMS, 171 (1972), 235–282.

    MathSciNet  MATH  Google Scholar 

  28. ——, “Reconstructing combinatorial geoemetries,” Graphs and Combinatorics, Springer-Verlag, Lecture Notes in Mathematics 406 (1974), 226–235.

    MathSciNet  Google Scholar 

  29. ——, “Modular constructions for combinatorial geometries,” Transactions of AMS, 203 (1975), 1–44.

    MathSciNet  MATH  Google Scholar 

  30. ——, “On the nonreconstructibility of combinatorial geometries,” Journal of Comb. Theory (B), 19 (1975), 72–76.

    MathSciNet  MATH  Google Scholar 

  31. Brylawski, T., “An Affine representation for transversal geometries,” Studies in Applied Mathematics, 54 (1975), 143–160.

    MathSciNet  MATH  Google Scholar 

  32. ——, “A Combinatorial perspective on the Radon convexity theorem”, Geometriae Dedicata, 5 (1976), 459–466.

    MathSciNet  MATH  Google Scholar 

  33. ——, “A Determinantal identity for resistive networks,” SIAM J. Appl. Math., 32 (1977), 443–449.

    MathSciNet  MATH  Google Scholar 

  34. ——, “Connected matroids with smallest Whitney numbers,” Discrete Math. 18 (1977), 243–252.

    MathSciNet  MATH  Google Scholar 

  35. ——, ”The Broken-circuit complex,” Transactions of AMS, 234 (1977), 417–433.

    MathSciNet  MATH  Google Scholar 

  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. ——, “Intersection theory for embeddings of matroids into uniform geometries,” Studies in Applied Mathematics 61 (1979), 211”244.

    MathSciNet  MATH  Google Scholar 

  38. ——, “The Affine dimension of the space of intersection matrices,” Rendiconti di Mathematics 13 (1980), 59–68.

    MathSciNet  MATH  Google Scholar 

  39. ——, “Intersection theory for graphs,” J. Comb. Th. (B) 30 (1981), 233–246.

    MathSciNet  MATH  Google Scholar 

  40. ——, “Hyperplane reconstruction of the Tutte polynomial of a geometric lattice,” Discrete Math. 35 (1981), 25–38.

    MathSciNet  MATH  Google Scholar 

  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. x2014;—, Matroids and Combinatorial Geometries, Carolina Lecture Series Volumn 8, Chapel Hill, N. C., 1980.

    MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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. Brylawski, T. and Oxley, J., “The Broken-circuit complex: its structure and factorizations,” European J. Combinatorics 2 (1981), 107–121.

    MathSciNet  MATH  Google Scholar 

  46. ——, “Several identities for the characteristic polynomial of a combinatorial geometry,” Discrete Math. 31 (1980), 161–170.

    MathSciNet  MATH  Google Scholar 

  47. Cardy, S., “The Proof of and generalisations to a conjecture by Baker and Essam,” Discrete Math. 4 (1973), 101–122.

    MathSciNet  MATH  Google Scholar 

  48. Cordovil, R., “Contributions à la théorie des géométries combinatories,” Thesis, 1'Université Pierre et Marie Curie, Paris, France.

    Google Scholar 

  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. Cordovil, R., Las Vergnas, M., and Mandel, A., “Euler's relation, Mobius functions, and matroid identities” (preprint, 1980).

    Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  52. Crapo, H. H., “The Mobius function of a lattice,” J. Comb. Th. 1 (1966), 126–131.

    MathSciNet  MATH  Google Scholar 

  53. x2014;—, “A Higher invariant for matroids,” J. Comb. Th. 2 (1967), 406–417.

    MathSciNet  MATH  Google Scholar 

  54. x2014;—, “Möbius inversion in lattices,” Archiv. der Math. 19 (1968), 595–607.

    MathSciNet  Google Scholar 

  55. ——, “The Joining of exchange geometries,” J. Math. Mech. 17 (1968), 837–852.

    MathSciNet  MATH  Google Scholar 

  56. ——, “The Tutte polynomial,” Aequationes Math. 3 (1969), 211–229.

    MathSciNet  MATH  Google Scholar 

  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. x2014;—, “Erecting geometries,” Proceedings of 2nd Chapel Hill Conference on Combinatorial Math. (1970), 74–99.

    Google Scholar 

  59. ——, “Constructions in combinatorial geometries,” (N.S.F. Advanced Science Seminar in Combinatorial Theory) (Notes, Bowdoin College), 1971).

    Google Scholar 

  60. Crapo, H. H. and Rota, G.-C, “On the Foundations of Combinatorial Theory: Combinatorial Geometries (preliminary edition), M.I.T. Press, 1970.

    MATH  Google Scholar 

  61. d'Antona, 0. and Kung, J. P. S., “Coherent orientations and series-parallel networks,” Disc. Math. 32 (1980), 95–98.

    MathSciNet  MATH  Google Scholar 

  62. Deza, M., “On perfect matroid designs,” Proc. Kyoto Conference, 1977, 98–108.

    Google Scholar 

  63. Deza, M. and Singi, N. M., “Some properties of perfect matroid designs,” Ann. Disc. Math. 6 (1980).

    Google Scholar 

  64. Dirac, G. A., “A roperty of 4-chromatic graphs and some remarks on critical graphs,” J. London Math. Soc. 27 (1952), 85–92.

    MathSciNet  MATH  Google Scholar 

  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. ——, “A Class of geometric lattices based on finite groups,” J. Comb. Th. 13, (1973), 61–87.

    MathSciNet  Google Scholar 

  67. x2014;—, “A q-analog of the partition lattice,” A Survey of Combinatorial Theory, North Holland (1973), 101–115.

    Google Scholar 

  68. Dowling, T. A. and Wilson, R. M., “The Slimmest geometric lattices,” Trans. Amer. Math. Soc. 196 (1974), 203–215.

    MathSciNet  MATH  Google Scholar 

  69. Edmonds, J. and Fulkerson, D. R., “Transversals and matroid partition,” J. Res. Nat. Bur. Stand. 69B (1965), 147–153.

    MathSciNet  Google Scholar 

  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. Essam, J. W., “Graph theory and statistical physics,” Discrete Math. 1 (1971), 83–112.

    MathSciNet  MATH  Google Scholar 

  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. 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.

    MathSciNet  MATH  Google Scholar 

  74. x2014;—, “On the Mobius algebra of a partially ordered set,” Advances in Math. 10 (1973), 177–187.

    MathSciNet  MATH  Google Scholar 

  75. ——, “Weight enumeration and the geometry of linear codes,” Studies in Appl. Math. 55 (1976), 119–128.

    MathSciNet  MATH  Google Scholar 

  76. ——, “Acyclic orientations,” (Notes), Higher Combinatorics, M. Aigner, ed., D. Reidel, Dordrecht (1977), 65–68.

    Google Scholar 

  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. Greenwell, D. L. and Hemminger, R. L., “Reconstructing graphs,” The Many Facets of Graph Theory, Springer-Verlag, Berlin, 1969, 91–114.

    Google Scholar 

  79. Hardy, G. H., Littlewood, J. E., and Pólya, G., Inequalities, Cambridge U. Press, 1934.

    Google Scholar 

  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. Hsieh, W. N. and Kleitman, D. J., “Normalized matching in direct products of partial orders,” Studies in Applied Math. 52 (1973), 285–289.

    MathSciNet  MATH  Google Scholar 

  82. ——, “Flows and generalized coloring theorems in graphs,” J. Comb. Th. (B) 26 (1979), 205–216.

    Google Scholar 

  83. ——, “A Constructive approach to the critical problem ” (to appear: Europ. J_. Combinatorics, 1981).

    Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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. Knuth, D. E., “The Asymptotic number of geometries,” J. Comb. Th. (A) 17 (1974), 398–401.

    Google Scholar 

  87. Las Vergnas, M., “Matroids orientables,” C. R. Acad. Sci. (Paris), 280A (1975), 61–64.

    MathSciNet  Google Scholar 

  88. ——, “Extensions normales d'un matroide, polynôme de Tutte d'un morphisme,” C. R. Acad. Sci. (Paris), 280 (1975), 1479–1482.

    MathSciNet  MATH  Google Scholar 

  89. ——, “Acyclic and totally cyclic orientations of combinatorial geometries,” Disc. Math., 20 (1977), 51–61.

    MathSciNet  Google Scholar 

  90. ——, “Sur les activités des orientations d'une geometrie combinatoire,” Collogue Mathématiques Discrètes: Codes et Hypergraphes, Bruxelles, 1978, 293–300.

    Google Scholar 

  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. 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. ——, “On the Tutte polynomial of a morphism of matroids,” Proc. Joint Canada-France Combinatorial Colloquium, Montréal 1979, Annals Discrete Math. 8 (1980), 7–20.

    MathSciNet  MATH  Google Scholar 

  94. Lindner, C. C. and Rosa, A., “Steiner quadruple systems a survey,” Discrete Math. 22:147–181 (1978).

    MathSciNet  MATH  Google Scholar 

  95. Lindström, B., “On the chromatic number of regular matroids,” J. Comb. Theory (B) 24 (1978), 367–369.

    MATH  Google Scholar 

  96. Lucas, T. D., “Properties of rank preserving weak maps,” A.M.S. Bull. 80 (1974), 127–131.

    MathSciNet  MATH  Google Scholar 

  97. ——, “Weak maps of combinatorial geometries,” Trans. Am. Math. Soc. 206 (1975), 247–279.

    MathSciNet  MATH  Google Scholar 

  98. Macwilliaras, F. J., “A Theorem on the distribution of weights in a systematic code,” Bell System Tech. J. 42 (1963), 79–94.

    MathSciNet  Google Scholar 

  99. Martin, P., “Enumerations eulériennes dans les multigraphes et invariants de Tutte-Grothendieck,” Thesis, Grenoble, 1977.

    Google Scholar 

  100. ——, “Remarkable valuation of the dichromatic polynomial of planar multigraphs,” J. Comb. Th. (B) 24 (1978), 318–324.

    MATH  Google Scholar 

  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. ——, “Matroids as the study of geometrical configurations,” Higher Combinatorics, M. Aigner, ed., D. Reidel, Dordrecht, Holland, 1977, 133–176.

    Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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. Hurty, U.S.R., “Equicardinal matroids,” J. Comb. Th. 11 (1971), 120–126.

    MATH  Google Scholar 

  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. Oxley, J. G., “Colouring, packing and the critical problem,” Quart. J. Math. Oxford, (2), 29, 11–22.

    Google Scholar 

  108. ——, “Cocircuit coverings and packings for binary matroids,” Math. Proc. Cambridge Philos. Soc. 83 (1978), 347–351.

    MathSciNet  MATH  Google Scholar 

  109. ——, “On cographic regular matroids,” Discrete Math. 25 (1979), 89–90.

    MathSciNet  MATH  Google Scholar 

  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. ——, “On a covering problem of Mullin and Stanton for binary matroids,” Aequationes Math. 19 (1979), 118, and 20 (1980), 104–112.

    MathSciNet  Google Scholar 

  112. ——, “On Crapo's beta invariant for matroids,” Studies in Appl. Math. (to appear).

    Google Scholar 

  113. ——, “On a matroid identity” (preprint, 1981).

    Google Scholar 

  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. Oxley, J. G. and Welsh, D. J. A., “On some percolation results of J. M. Hammersley,” J. Appl. Probability 16 (1979), 526–540.

    MathSciNet  MATH  Google Scholar 

  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. Read, R. C, “An Introduction to chromatic polynomials,” J. Comb. Th., 4 (1968), 52–71.

    MathSciNet  Google Scholar 

  118. Rota, G.-C, “On the foundations of combinatorial theory I,” Z. Wahrsch, 2 (1964), 340–368.

    MathSciNet  MATH  Google Scholar 

  119. ——, “Combinatorial analysis as a theory,” Hedrick Lectures, Math. Assoc, of Amer., Summer Meeting, Toronto, 1967.

    Google Scholar 

  120. ——, “Combinatorial theory, old and new,“ Int. Cong. Math. (Nice) (1970) 3, 229–233.

    Google Scholar 

  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. ——, “Calotte di tipo (m,n) in uno spazio di Galois sr,q,” Rend. Acc. Naz. Lincei 53(8) (1973), 71–81.

    MATH  Google Scholar 

  123. Segre, B., Lectures on Modern Geometry, Edizioni Creomonese, Roma, 1961.

    MATH  Google Scholar 

  124. Seymour, P. D., “On Tutte's extension of the four-colour problem ” (preprint, 1979).

    Google Scholar 

  125. ——, “Decomposition of regular matroids,” J. Comb. Th. (B) 28 (1980), 305–359.

    Google Scholar 

  126. ——, “Nowhere-zero 6-flows,” J. Comb. Th. (B) 30 (1981), 130–135.

    Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  128. Shepherd, G. C., “Combinatorial properties of associated zonotopes,” Can. J. Math. 26 (1974), 302–321.

    Google Scholar 

  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. ——, “Patroids,” J. Comb. Th. 16 (1974), 64–76.

    Google Scholar 

  131. Stanley, R., “Modular elements of geometric lattices,” Algebra Universalis, 1 (1971), 214–217.

    MathSciNet  MATH  Google Scholar 

  132. ——, “Supersolvable semimodular lattices,” Proc. Conference on Möbius Algebras, University of Waterloo, 1971, pp. 80–142.

    Google Scholar 

  133. ——, “Supersolvable lattices,” Alg. Universalis 2 (1972), 197–217.

    MathSciNet  MATH  Google Scholar 

  134. ——, “Acyclic orientations of graphs,” Disc. Math. 5 (1974), 171–178.

    MathSciNet  Google Scholar 

  135. Szekeres, G. and Wilf, H., “An Inequality for the chromatic number of a graph,” J. Comb. Th. 4 (1968), 1–3.

    MathSciNet  Google Scholar 

  136. Tallini, G., “Problemi e risultati sulle geometrie di Galois,” Rel. N. 30, 1st. di Mat. dell' Univ. di Napoli (1973).

    Google Scholar 

  137. Tutte, W. T., “A Ring in graph theory,” Proc. Cambridge Phil Soc. 43 (1947), 26–40.

    MathSciNet  MATH  Google Scholar 

  138. ——, “A Contribution to the theory of chromatic polynomials,” Canad. J. Math. 6 (1954), 80–91.

    MathSciNet  MATH  Google Scholar 

  139. ——, “A Class of Abelian groups,” Canad. J. Math. 8 (1956), 13–28.

    MathSciNet  MATH  Google Scholar 

  140. ——, “Matroids and graphs,” Trans. Amer. Math. Soc. 90 (1959), 527–552.

    MathSciNet  MATH  Google Scholar 

  141. ——, “Lectures on matroids,” J. Res. Nat. Bur. Stand. 69B (1965), 1–48.

    MathSciNet  Google Scholar 

  142. ——, “On the algebraic theory of graph coloring,” J. Comb. Th. 1 (1966), 15–50.

    MathSciNet  MATH  Google Scholar 

  143. ——, “On dichromatic polynomials,” J. Comb. Th. 2(1967), 301–320.

    MathSciNet  MATH  Google Scholar 

  144. ——, “Projective geometry and the 4-color problem,” Recent Progress in Combinatorics (W. T. Tutte, ed.) Academic Press 1969, 199–207.

    Google Scholar 

  145. ——, “Codichromatic graphs,” J. Comb. Th. 16 (1974), 168–175.

    MathSciNet  MATH  Google Scholar 

  146. ——, “All the king's men (a guide to reconstruction),” Graph Theory and Related Topics, Academic Press, 1979, 15–33.

    Google Scholar 

  147. Van Lint, J. H., Coding Theory, Springer Lecture Notes, 201, (1971).

    Google Scholar 

  148. Walton, P. N. and Welsh, D. J. A., “On the chromatic number of binary matroids,” Mathematika 27 (1980), 1–9.

    MathSciNet  MATH  Google Scholar 

  149. Welsh, D. J. A., “Euler and bipartite matroids,” J. Comb. Th. 6 (1969), 375–377.

    MathSciNet  MATH  Google Scholar 

  150. ——, “Combinatorial problems in matroid theory,” Combinatorial Mathematics and its Applications, Academic Press, (1971), 291–307.

    Google Scholar 

  151. ——, Matroid Theory, Academic Press, London, 1976.

    MATH  Google Scholar 

  152. ——, “Percolation and related topoics,” Science Progress 64 (1977).

    Google Scholar 

  153. ——, “Colouring problems and matroids,“ Proc. Seventh British Combinatorial Conference, Cambridge U. Press (1979), 229–257.

    Google Scholar 

  154. Welsh, D. J. A., “Colourings, flows and projective geometry,” Nieuw Archief voor Wiskunde (3), 28 (1980), 159–176.

    MathSciNet  MATH  Google Scholar 

  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. Whitney, H., “A Logical expansion in mathematics,” Bull. Amer. Math. Soc. 38 (1932), 572–579.

    MathSciNet  Google Scholar 

  157. ——, “The Coloring of graphs,” Annals of Math. 33 (1932), 688–718.

    MathSciNet  Google Scholar 

  158. ——, “2-isomorphic graphs,” Amer. J. Math. 55 (1933), 245–254.

    MathSciNet  Google Scholar 

  159. ——, “On the abstract properties of linear dependence,” Amer. J. Math. 57 (1935), 509–533.

    MathSciNet  Google Scholar 

  160. Wilf, H. S., “Which polynomials are chromatic?” Atti dei Convegni Lincei 17, Tomo 1 (1976), 247–256.

    Google Scholar 

  161. Winder, R. O., “Partitions of n-space by hyperplanes,” SlAM J. Appl. Math. 14 (1966), 811–818.

    MathSciNet  MATH  Google Scholar 

  162. Young, P. and Edmonds, J., “Matroid designs,” J. Res. Nat. Bur. Stan. 72B (1972), 15–44.

    Google Scholar 

  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. ——, “Counting faces of cut-up spaces,“ Bull. Amer. Math. Soc. 81 (1975), 916–918.

    MathSciNet  MATH  Google Scholar 

  165. ——, “Maximal dissections of a simplex,” J. Comb. Th. (A) 20 (1976), 244–257.

    MathSciNet  MATH  Google Scholar 

  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. ——, “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.

    MathSciNet  Google Scholar 

  168. ——, “The Geometry of root systems and signed graphs,” Amer. Math. Monthly, 88 (1981), 88–105.

    MathSciNet  MATH  Google Scholar 

  169. Zaslavsky, T., “Signed graphs” (preprint, 1980).

    Google Scholar 

  170. ——, “Orientation of signed graphs” (preprint, 1980).

    Google Scholar 

  171. ——, “Signed graph coloring” (preprint, 1980).

    Google Scholar 

  172. ——, “Chromatic invariants of signed graphs” (preprint, 1980).

    Google Scholar 

  173. ——, “Bicircular geometry and the lattice of forest of a graph” (preprint, 1980).

    Google Scholar 

  174. ——, “The slimmest arrangements of hyperplanes: I. Geometric lattices and projective arrangements” (preprint, 1980).

    Google Scholar 

  175. ——, “The slimmest arrangements of hyperplanes: II. Basepointed geometric lattices and Euclidean arrangements (preprint, 1980).

    Google Scholar 

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  1. Department of Mathematics, University of North Carolina, Chapel Hill, N.C., 27514, USA

    Thomas Brylawski

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A. Barlotti

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Brylawski, T. (2010). The Tutte Polynomial Part I: General Theory. In: Barlotti, A. (eds) Matroid Theory and its Applications. C.I.M.E. Summer Schools, vol 83. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11110-5_3

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