# The Tutte Polynomial Part I: General Theory

• Thomas Brylawski
Chapter
Part of the C.I.M.E. Summer Schools book series (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
$${\text{t}}\left( {{\text{M;}}\,{\text{x, y}}} \right) = \sum {{\text{a}}_{{\text{ij}}} \left( {{\text{x - 1}}} \right)^{\text{i}} \left( {{\text{y - 1}}} \right)^{\text{j}} }$$

## Preview

### Bibliography

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.
3. 3.
——, “The Möbius algebra as a Grothendieck ring,” J. of Algebra 57 (1979), 167–179.
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.
11. 11.
Birkhoff, G. D. and Lewis, D. C., “Chromatic polynomials,” Trans. Amer. Math. Soc. 60 (1946), 355–451.
12. 12.
Bixby, R. E., “A omposition for matroids,” J. Comb. Th. (B) 18 (1975), 59–73.
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.
16. 16.
Bland, R. G. and Las Vergnas, M., “Orientability of matroids,” J. Comb. Th. (B) 24 (1978), 94–123.
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.
20. 20.
Brooks, R. L., “On colouring the nodes of a network,” Proc. Cambridge Phil. Soc. 37 (1941), 194–197.
21. 21.
Brouwer, A. E. and Schriver, A., “The blocking number of an affine space,” J. Comb. Th. (A) 24 (1978), 251–253.
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.
24. 24.
Brylawski, T., “A Combinatorial model for series-parallel networks,” Transactions of the AMS, 154 (1971), 1–22.
25. 25.
——, “Some properties of basic families of subsets,” Disc. Math. 6 (1973), 333–341.
26. 26.
——, “The Tutte-Grothendieck ring,” Algebra Universalis 2 (1972), 375–388.
27. 27.
——, “A Decomposition for combinatorial geometries,” Transactions of the AMS, 171 (1972), 235–282.
28. 28.
——, “Reconstructing combinatorial geoemetries,” Graphs and Combinatorics, Springer-Verlag, Lecture Notes in Mathematics 406 (1974), 226–235.
29. 29.
——, “Modular constructions for combinatorial geometries,” Transactions of AMS, 203 (1975), 1–44.
30. 30.
——, “On the nonreconstructibility of combinatorial geometries,” Journal of Comb. Theory (B), 19 (1975), 72–76.
31. 31.
Brylawski, T., “An Affine representation for transversal geometries,” Studies in Applied Mathematics, 54 (1975), 143–160.
32. 32.
——, “A Combinatorial perspective on the Radon convexity theorem”, Geometriae Dedicata, 5 (1976), 459–466.
33. 33.
——, “A Determinantal identity for resistive networks,” SIAM J. Appl. Math., 32 (1977), 443–449.
34. 34.
——, “Connected matroids with smallest Whitney numbers,” Discrete Math. 18 (1977), 243–252.
35. 35.
——, ”The Broken-circuit complex,” Transactions of AMS, 234 (1977), 417–433.
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.
38. 38.
——, “The Affine dimension of the space of intersection matrices,” Rendiconti di Mathematics 13 (1980), 59–68.
39. 39.
——, “Intersection theory for graphs,” J. Comb. Th. (B) 30 (1981), 233–246.
40. 40.
——, “Hyperplane reconstruction of the Tutte polynomial of a geometric lattice,” Discrete Math. 35 (1981), 25–38.
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.
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.
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.
46. 46.
——, “Several identities for the characteristic polynomial of a combinatorial geometry,” Discrete Math. 31 (1980), 161–170.
47. 47.
Cardy, S., “The Proof of and generalisations to a conjecture by Baker and Essam,” Discrete Math. 4 (1973), 101–122.
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.
52. 52.
Crapo, H. H., “The Mobius function of a lattice,” J. Comb. Th. 1 (1966), 126–131.
53. 53.
x2014;—, “A Higher invariant for matroids,” J. Comb. Th. 2 (1967), 406–417.
54. 54.
x2014;—, “Möbius inversion in lattices,” Archiv. der Math. 19 (1968), 595–607.
55. 55.
——, “The Joining of exchange geometries,” J. Math. Mech. 17 (1968), 837–852.
56. 56.
——, “The Tutte polynomial,” Aequationes Math. 3 (1969), 211–229.
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.
61. 61.
d'Antona, 0. and Kung, J. P. S., “Coherent orientations and series-parallel networks,” Disc. Math. 32 (1980), 95–98.
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.
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.
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.
69. 69.
Edmonds, J. and Fulkerson, D. R., “Transversals and matroid partition,” J. Res. Nat. Bur. Stand. 69B (1965), 147–153.
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.
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.
74. 74.
x2014;—, “On the Mobius algebra of a partially ordered set,” Advances in Math. 10 (1973), 177–187.
75. 75.
——, “Weight enumeration and the geometry of linear codes,” Studies in Appl. Math. 55 (1976), 119–128.
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.
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.
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.
88. 88.
——, “Extensions normales d'un matroide, polynôme de Tutte d'un morphisme,” C. R. Acad. Sci. (Paris), 280 (1975), 1479–1482.
89. 89.
——, “Acyclic and totally cyclic orientations of combinatorial geometries,” Disc. Math., 20 (1977), 51–61.
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.
94. 94.
Lindner, C. C. and Rosa, A., “Steiner quadruple systems a survey,” Discrete Math. 22:147–181 (1978).
95. 95.
Lindström, B., “On the chromatic number of regular matroids,” J. Comb. Theory (B) 24 (1978), 367–369.
96. 96.
Lucas, T. D., “Properties of rank preserving weak maps,” A.M.S. Bull. 80 (1974), 127–131.
97. 97.
——, “Weak maps of combinatorial geometries,” Trans. Am. Math. Soc. 206 (1975), 247–279.
98. 98.
Macwilliaras, F. J., “A Theorem on the distribution of weights in a systematic code,” Bell System Tech. J. 42 (1963), 79–94.
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.
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.
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.
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.
109. 109.
——, “On cographic regular matroids,” Discrete Math. 25 (1979), 89–90.
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.
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.
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.
118. 118.
Rota, G.-C, “On the foundations of combinatorial theory I,” Z. Wahrsch, 2 (1964), 340–368.
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.
123. 123.
Segre, B., Lectures on Modern Geometry, Edizioni Creomonese, Roma, 1961.
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.
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.
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.
134. 134.
——, “Acyclic orientations of graphs,” Disc. Math. 5 (1974), 171–178.
135. 135.
Szekeres, G. and Wilf, H., “An Inequality for the chromatic number of a graph,” J. Comb. Th. 4 (1968), 1–3.
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.
138. 138.
——, “A Contribution to the theory of chromatic polynomials,” Canad. J. Math. 6 (1954), 80–91.
139. 139.
——, “A Class of Abelian groups,” Canad. J. Math. 8 (1956), 13–28.
140. 140.
——, “Matroids and graphs,” Trans. Amer. Math. Soc. 90 (1959), 527–552.
141. 141.
——, “Lectures on matroids,” J. Res. Nat. Bur. Stand. 69B (1965), 1–48.
142. 142.
——, “On the algebraic theory of graph coloring,” J. Comb. Th. 1 (1966), 15–50.
143. 143.
——, “On dichromatic polynomials,” J. Comb. Th. 2(1967), 301–320.
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.
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.
149. 149.
Welsh, D. J. A., “Euler and bipartite matroids,” J. Comb. Th. 6 (1969), 375–377.
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.
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.
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.
157. 157.
——, “The Coloring of graphs,” Annals of Math. 33 (1932), 688–718.
158. 158.
——, “2-isomorphic graphs,” Amer. J. Math. 55 (1933), 245–254.
159. 159.
——, “On the abstract properties of linear dependence,” Amer. J. Math. 57 (1935), 509–533.
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.
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.
165. 165.
——, “Maximal dissections of a simplex,” J. Comb. Th. (A) 20 (1976), 244–257.
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.
168. 168.
——, “The Geometry of root systems and signed graphs,” Amer. Math. Monthly, 88 (1981), 88–105.
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