A Tutte Polynomial for Coloured Graphs

Combinatorics Probability and Computing

Volumn 8, Issue 1-2, 1999, Pages 45-93

  Bollobás, Béla ^a,b,c   Riordan, Oliver ^a,b  

^a University of Memphis   (United States)

^b UNIVERSITY OF CAMBRIDGE   (United Kingdom)

^c INSTITUTE FOR ADVANCED STUDY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0002601708     PISSN: 09635483     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0963548398003447     Document Type: Article

References (40)

1. Alon, N., Frieze, A. and Welsh, D. J. A. (1995) Polynomial time randomized approximation schemes for Tutte-Grothendieck invariants: the dense case. Random Structures and Algorithms 6 459-478.
2. Annan, J. D. (1995) The complexities of the coefficients of the Tutte polynomial. Discrete Appl. Math. 57 93-103.
3. Birkhoff, G. D. (1912) A determinant formula for the number of ways of coloring a map. Ann. Math. 14 42-46.
4. Brylawski, T. and Oxley, J. (1992) The Tutte polynomial and its applications. In Matroid Applications (N. White, ed.), Cambridge University Press, pp. 123-225.
5. Burde, G. and Zieschang, H. (1985) Knots, De Gruyter, Berlin.
6. de la Harpe, P., Kervaire, M. and Weber, C. (1986) On the Jones polynomial. L'Enseignement Mathématique 32 271-335.
7. Fortuin, C. M. and Kasteleyn, P. W. (1972) On the random-cluster model: I. Physica 57 536-564.
8. Freyd, P., Yetter, D., Hoste, J., Lickorish, W. B. R., Millett K. and Ocneanu, A. (1985) A new polynomial invariant of knots and links. Bull. Amer. Math. Soc. 12 239-246.
9. Jaeger, F. (1988) Tutte polynomials and link polynomials. Proc. Amer. Math. Soc. 103 647-654.
10. Jaeger, F., Vertigan, D. L. and Welsh, D. J. A. (1990) On the computational complexity of the Jones and Tutte polynomials. Math. Proc. Cam. Phil. Soc. 108 35-53.
11. Jones, V. F. R. (1985) A polynomial invariant of knots via von Neumann algebras. Bull. Amer. Math. Soc. 12 103-111.
12. Kandri-Rody, A. and Kapur, D. (1984) Algorithms for computing Gröbner bases of polynomial ideals over various Euclidean rings. In EUROSAM 84 (Cambridge, 1984), Vol. 174 of Lecture Notes in Computer Science, Springer, Berlin-New York, pp. 195-206.
13. Kauffman, L. H. (1987) State models and the Jones polynomial. Topology 26 395-407.
14. Kauffman, L. H. (1989) A Tutte polynomial for signed graphs. Discrete Appl. Math. 25 105-127.
15. Kauffman, L. H. (1990) An invariant of regular isotopy. Trans. Amer. Math. Soc. 318 417-471.
16. Lickorish, W. B. R. (1988) Polynomials for links. Bull. London Math. Soc. 20 558-588.
17. Lipson, A. S. (1988) A note on some link polynomials. Bull. London Math. Soc. 20 532-534.
18. Lipson, A. S. (1992) Some more states models for link invariants, Pacific J. Math. 152 337-346.
19. Murasugi, K. (1987) Jones polynomials and classical conjectures in knot theory. Topology 26 187-194.
20. Murasugi, K. (1989) On invariants of graphs with applications to knot theory. Trans. Amer. Math. Soc. 314 1-49.
21. Negami, S. (1987) Polynomial invariants of graphs. Trans. Amer. Math. Soc. 299 601-622.
22. Oxley, J. G. and Welsh, D. J. A. (1979) The Tutte polynomial and percolation. In Graph Theory and Related Topics (J. A. Bondy and U. S. R. Murty, eds), Academic Press, New York, pp. 329-339.
23. Przytycki, J. H. and Traczyk, P. (1987) Invariants of links of Conway type. Kobe J. Math. 4 115-139.
24. Reidemeister, K. (1948) Knotentheorie, Chelsea Publishing Co., New York. Copyright 1932, Julius Springer, Berlin.
25. Rolfsen, D. (1976) Knots and Links, Publish or Perish, Berkeley, 439pp.
26. Schwärzler, W. and Welsh, D. J. A. (1993) Knots, matroids and the Ising model. Math. Proc. Cam. Phil. Soc. 113 107-139.
27. Sekine, K., Imai, H. and Tani, S. (1995) Computing the Tutte polynomial of a graph of moderate size. In Algorithms and Computations, Vol. 1004 of Lecture Notes in Computer Science, Springer, Berlin, pp. 224-233.
28. Tait, P. G. (1898) On knots I, II, III. In Scientific Papers, Vol. I, Cambridge University Press, pp. 273-347.
29. Thistlethwaite, M. B. (1987) A spanning tree expansion of the Jones polynomial. Topology 26 297-309.
30. Thistlethwaite, M. B. (1988) Kauffman's polynomial and alternating links. Topology 27 311-318.
31. Thistlethwaite, M. B. (1988) On the Kauffman polynomial of an adequate link. Invent. Math. 93 285-296.
32. Traldi, L. (1989) A dichromatic polynomial for weighted graphs and link polynomials. Proc. Amer. Math. Soc. 106 279-286.
33. Tutte, W. T. (1947) A ring in graph theory. Proc. Cam. Phil. Soc. 43 26-40.
34. Tutte, W. T. (1954) A contribution to the theory of chromatic polynomials. Canadian J. Math. 6 80-91.
35. Tutte, W. T. (1967) On dichromatic polynomials. J. Combin. Theory 2 301-320.
36. Welsh, D. J. A. (1993) Complexity: Knots, Colourings and Counting, Vol. 186 of London Math. Soc. Lecture Notes, Cambridge University Press.
37. Welsh, D. J. A. (1995) Randomised approximation schemes for Tutte-Grothendieck invariants, in Discrete Probability and Algorithms, Vol. 72 of IMA Vol. Math. Appl., Springer, New York, pp. 133-148.
38. Whitney, H. (1932) A logical expansion in mathematics. Bull. Amer. Math. Soc. 38 572-579.
39. Whitney, H. (1932) The coloring of graphs. Ann. Math. 33 688-718.
40. Zaslavsky, T. (1992) Strong Tutte functions of matroids and graphs. Trans. Amer. Math. Soc. 334 317-347.