Hostname: page-component-6bf8c574d5-k2jvg Total loading time: 0 Render date: 2025-03-09T14:30:24.879Z Has data issue: false hasContentIssue false
  • English
  • Français

A Contribution to the Theory of Chromatic Polynomials

Published online by Cambridge University Press:  20 November 2018

W. T. Tutte
W. T. Tutte*
Affiliation:
University of Toronto
Rights & Permissions [Opens in a new window]

Summary

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Two polynomials θ(G, n) and ϕ(G, n) connected with the colourings of a graph G or of associated maps are discussed. A result believed to be new is proved for the lesser-known polynomial ϕ(G, n). Attention is called to some unsolved problems concerning ϕ(G, n) which are natural generalizations of the Four Colour Problem from planar graphs to general graphs. A polynomial χ(G, x, y) in two variables x and y, which can be regarded as generalizing both θ(G, n) and ϕ(G, n) is studied. For a connected graph χ(G, x, y) is defined in terms of the “spanning” trees of G (which include every vertex) and in terms of a fixed enumeration of the edges.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 6 , 1954 , pp. 80 - 91
Copyright
Copyright © Canadian Mathematical Society 1954

References

1. Birkhoff, G. D. and Lewis, D. C., Chromatic polynomials, Trans. Amer. Math. Soc, 60 (1946), 355–451.Google Scholar
2. Brooks, R. L., Smith, C. A. B., Stone, A. H. and Tutte, W. T., The dissection of rectangles into squares, Duke Math. J., 7 (1940), 312–340.Google Scholar
3. König, Denes, Theorie der endlichen una unendlichen Graphen (Leipzig, 1936).Google Scholar
4. Petersen, J., Sur le théorème de Tait, L'Intermédiaire des mathématiciens, 5, (1898), 225–227.Google Scholar
5. Tutte, W. T., On the imbedding of linear graphs in surfaces, Proc. London Math. Soc. (2), 61 (1949), 474–483.Google Scholar
6. Whitney, Hassler, A logical expansion in mathematics, Bull. Amer. Math. Soc, 88 (1932), 572–579.Google Scholar
7. Whitney, Hassler, The coloring of graphs, Ann. Math., 33 (1932), 688–718.Google Scholar