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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
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.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1975 Springer-Verlag
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/BFb0069561
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-07154-9
Online ISBN: 978-3-540-37482-4
eBook Packages: Springer Book Archive