Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T21:08:06.787Z Has data issue: false hasContentIssue false
  • English
  • Français

On the chromatic number of binary matroids

Part of: Designs and configurations

Published online by Cambridge University Press:  26 February 2010

P. N. Walton  and
D. J. A. Welsh
P. N. Walton
Affiliation:
Merton College Oxford.
D. J. A. Welsh
Affiliation:
Merton College Oxford.
Get access

Extract

In this paper we obtain matroid extensions of two important results in graph theory, namely the 4-colour theorem of Appel and Haken [1] and the 8-flow theorem of Jaeger [4]. As a corollary we prove that any bridgeless graph with no subgraph contractible to K3,3 has a nowhere zero 4-flow. These results depend heavily on a remarkable theory of splitters developed recently by Seymour [8], [9].

MSC classification

Secondary: 05B35: Matroids, geometric lattices
Type
Research Article
Information
Mathematika , Volume 27 , Issue 1 , June 1980 , pp. 1 - 9
Copyright
Copyright © University College London 1980

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Appel, K. and Haken, W.. “Every planar map is four-colourable”, Bull. Amer. Math. Soc., 82 (1976), 711–712.CrossRefGoogle Scholar
2.Bondy, J. A. and Murty, U. S. R.. Graph Theory with Applications (Macmillan, 1976).CrossRefGoogle Scholar
3.Brylawski, T. H.. “Modular constructions for combinatorial geometries”, Trans. Amer. Math. Soc., 203 (1975), 1–44.CrossRefGoogle Scholar
4.Jaeger, F.. “On nowhere zero flows in multigraphs”, Proc. Fifth British Comb. Conf. ed. Nash-Williams, C. St. J. A. and Sheehan, J. (Utilitas, 1976), 373–379.Google Scholar
5.Jaeger, F.. “Flows and generalised colouring theorems in graphs”, J. Comb. Theory B, (to appear).Google Scholar
6.Lindström, B.. “On the chromatic number of regular matroids”, J. Comb. Theory B, 24 (1978), 367–369.CrossRefGoogle Scholar
7.Oxley, J. G.. “Colouring, packing and the critical problem”, Quart. J. Math. Oxford, (2), 29 (1978), 11–22.CrossRefGoogle Scholar
8.Seymour, P. D.. “On Tutte's extension of the four colour problem” (preprint, 1979).Google Scholar
9.Seymour, P. D.. “Decomposition of regular matroids”, J. Comb. Theory (1979) (to appear).Google Scholar
10.Tutte, W. T.. “A contribution to the theory of chromatic polynomials”, Canad. J. Math., 6 (1954), 80–91.CrossRefGoogle Scholar
11.Tutte, W. T.. “Lectures on matroids”, J. Res. Nat. Bur. Stand., 69B (1965), 1–18.Google Scholar
12.Wagner, K.. “Beweis einer Abschwàchung der Hadwiger-Vermutung”, Math. Ann., 153 (1964), 139–141.CrossRefGoogle Scholar
13.Welsh, D. J. A.. Matroid Theory, London Math. Soc. Monograph No. 8 (Academic Press, 1976).Google Scholar
14.Welsh, D. J. A.. “Colouring problems and matroids”, Proceedings of Seventh British Combinatorial Conference (Cambridge University Press, 1979), 229–257.Google Scholar