Short Proof of a Map-Colour Theorem

G. A. Dirac

doi:10.4153/CJM-1957-027-2

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Heawood (3) showed that for h > 1 the countries of any map on a surface of connectivity h = 3 − x can be coloured using at most

colours. In a previous paper (1)1 proved the following

References

1. Dirac, G. A., Map-colour theorems, Can. J. Math., 4 (1952), 480–490.Google Scholar

2. Dirac, G. A., A theorem of R. L. Brooks and a conjecture of H. Hadwiger, Proc. London Math. Soc. (3), 7 (1957).Google Scholar

3. Heawood, P. J., Map-colour theorem, Quart. J. Math., 24 (1890), 332–338.Google Scholar

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Kainen, Paul C. 1974. Graphs and Combinatorics. Vol. 406, Issue. , p. 76.

Albertson, Michael O. and Hutchinson, Joan P. 1977. The independence ratio and genus of a graph. Transactions of the American Mathematical Society, Vol. 226, Issue. 0, p. 161.

Albertson, Michael O. and Hutchinson, Joan P. 1979. THE THREE EXCLUDED CASES OF DIRAC'S MAP‐COLOR THEOREM1. Annals of the New York Academy of Sciences, Vol. 319, Issue. 1, p. 7.

Albertson, Michael O. and Hutchinson, Joan P. 1980. Hadwiger's conjecture for graphs on the Klein bottle. Discrete Mathematics, Vol. 29, Issue. 1, p. 1.

Albertson, Michael O. and Stromquist, Walter R. 1982. Locally planar toroidal graphs are 5-colorable. Proceedings of the American Mathematical Society, Vol. 84, Issue. 3, p. 449.

1988. Graph Theory in Memory of G.A. Dirac. Vol. 41, Issue. , p. 1.

Hartsfield, Nora and Ringel, Gerhard 1991. Clean triangulations. Combinatorica, Vol. 11, Issue. 2, p. 145.

Mohar, Bojan 1993. 7-critical graphs of bounded genus. Discrete Mathematics, Vol. 112, Issue. 1-3, p. 279.

1994. Graph Coloring Problems. p. 1.

1994. Graph Coloring Problems. p. 59.

Albertson, Michael O. and Haas, Ruth 1996. Parsimonious edge coloring. Discrete Mathematics, Vol. 148, Issue. 1-3, p. 1.

B�hme, T. Mohar, B. and Stiebitz, M. 1999. Dirac's map-color theorem for choosability. Journal of Graph Theory, Vol. 32, Issue. 4, p. 327.

Archdeacon, Dan Hutchinson, Joan Nakamoto, Atsuhiro Negam, Seiya and Ota, Katsuhiro 2001. Chromatic numbers of quadrangulations on closed surfaces. Journal of Graph Theory, Vol. 37, Issue. 2, p. 100.

Füredi, Zoltan Kostochka, Alexandr V. Škrekovski, Riste Stiebitz, Michael and West, Douglas B. 2005. Nordhaus–Gaddum‐type Theorems for decompositions into many parts. Journal of Graph Theory, Vol. 50, Issue. 4, p. 273.

Král', Daniel and S̆krekovski, Riste 2006. The last excluded case of Dirac's map‐color theorem for choosability. Journal of Graph Theory, Vol. 51, Issue. 4, p. 319.

Stiebitz, Michael and Škrekovski, Riste 2006. A map colour theorem for the union of graphs. Journal of Combinatorial Theory, Series B, Vol. 96, Issue. 1, p. 20.

Hutchinson, Joan P. 2016. Graph Theory. p. 195.

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