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Polyhedral decompositions of cubic graphs

Published online by Cambridge University Press:  17 April 2009

G. Szekeres
G. Szekeres
Affiliation:
School of Mathematics, University of New South Wales, Kensington, New South Wales.
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Abstract

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A polyhedral decomposition of a finite trivalent graph G is defined as a set of circuits = {C1, C2, … Cm} with the property that every edge of G occurs exactly twice as an edge of some Ck. The decomposition is called even if every Ck is a simple circuit of even length. If G has a Tait colouring by three colours a, b, c then the (a, b), (b, c) and (c, a) circuits obviously form an even polyhedral decomposition. It is shown that the converse is also true: if G has an even polyhedral decomposition then it also has a Tait colouring. This permits an equivalent formulation of the four colour conjecture (and a much stronger conjecture of Branko Grünbaum) in terms of polyhedral decompositions alone.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 8 , Issue 3 , June 1973 , pp. 367 - 387
Copyright
Copyright © Australian Mathematical Society 1973

References

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