Polyhedral decompositions of cubic graphs

Abstract

A polyhedral decomposition of a finite trivalent graphGis defined as a set of circuits= {C1, C2, …Cm} with the property that every edge ofGoccurs exactly twice as an edge of someCk. The decomposition is called even if everyCkis a simple circuit of even length. IfGhas a Tait colouring by three coloursa, b, cthen the (a, b), (b, c) and (c, a) circuits obviously form an even polyhedral decomposition. It is shown that the converse is also true: ifGhas 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.