Abelian Colourings of Cubic Graphs

Abstract

Abstract We prove that every bridgeless cubic graph G can have its edges properly coloured by non-zero elements of any given Abelian group A of order at least 12 in such a way that at each vertex of G the three colours sum to zero in A. The proof relies on the fact that such colourings depend on certain configurations in Steiner triple systems. In contrast, a similar statement for cyclic groups of order smaller than 10 is false, leaving the problem open only for Z 4 × Z 2 , Z 3 × Z 3 , Z 10 and Z 11 . All the extant cases are closely related to certain conjectures concerning cubic graphs, most notably to the celebrated Berge-Fulkerson Conjecture.