Abstract
Rufus Isaacs has recently constructed an infinite family of trivalent graphs, generalizations of the Petersen graph, which cannot be edge-coloured by three colours. In this paper it is shown that none of these graphs can form the boundary of a proper map (that is one whose dual is a triangulation) on an orientable surface. The result strengthens a conjecture of Branko Grunbaum (generalization of the four colour conjecture) that the dual of a triangulation on an orientable surface can always be Tait coloured.
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References
Rufus Isaacs, Infinite families of non-trivial trivalent graphs which are not Tait colorable, Technical Report No. 177, John Hopkins University (1974); Amer. Math. Monthly (to appear)
G. Szekeres, Polyhedral decompositions of cubic graphs, Bull. Austral. Math. Soc. 8 (1973) 367–387.
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© 1975 Springer-Verlag
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Szekeres, G. (1975). Non-colourable trivalent graphs. In: Street, A.P., Wallis, W.D. (eds) Combinatorial Mathematics III. Lecture Notes in Mathematics, vol 452. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0069561
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DOI: https://doi.org/10.1007/BFb0069561
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