Balanced colourings and the four colour conjecture

A conjectured property of bridgeless cubic planar graphs is shown to be equivalent to the four colour conjecture. In establishing this equivalence use is made of the Kbnig.Hall theorem on the existence of one-factors in bipartite graphs. 1. The four colour conjecture. For a discussion of the four colour conjecture and related topics we refer the reader to Ore [3]. We merely content ourselves here with a statement of the conjecture and of two equivalent conjectures (proofs of their equivalence are given in [3]). Our graph-theoretic terminology is that of Harary [2] (although we use vertices and edges for what are respectively called points and lines in [2]). FouR COLOUR CONJECTURE. The faces of any plane graph can be coloured with four colours so that no two incident faces are assigned the same colour. THE HEAWOOD CONJECTURE. For any bridgeless cubic plane graph G there is a mappingf: V(G)-3{1, + 1} so that, for every face F, J,Ff (x)0 (mod 3). THE TAIT CONJECTURE. Every bridgeless cubic planar graph has an even two-factor (that is a two-factor in which each cycle is of even length). 2. Balanced colourings. Let G be a graph with vertex set V(G) and edge set E(G). A partition W= (B, W) of V(G) is a two-colouring of V(G); ' is an equitable two-colouring if IBI = I WI. For a two-colouring ' and a vertex x, we define w(x, '), the weight of x in ', by w(x,jW=-2 if xeB, = +2 if xeW, and for Sc V(G) we denote 2ies w(x, ') by w(S, s). Again, for SC V(G), let v(S) denote the number of edges of G having exactly one end in S. A balanced colouring of G is a two-colouring ' of Received by the editors August 2, 1971. AMS 1970 subject classifications. Primary 05C15.