Even-cycle decompositions of graphs with no odd-K4-minor

Abstract An even-cycle decomposition of a graph G is a partition of E ( G ) into cycles of even length. Evidently, every Eulerian bipartite graph has an even-cycle decomposition. Seymour (1981) proved that every 2-connected loopless Eulerian planar graph with an even number of edges also admits an even-cycle decomposition. Later, Zhang (1994) generalized this to graphs with no K 5 -minor. Our main theorem gives sufficient conditions for the existence of even-cycle decompositions of graphs in the absence of odd minors . Namely, we prove that every 2-connected loopless Eulerian odd- K 4 -minor-free graph with an even number of edges has an even-cycle decomposition. This is best possible in the sense that ‘odd- K 4 -minor-free’ cannot be replaced with ‘odd- K 5 -minor-free.’ The main technical ingredient is a structural characterization of the class of odd- K 4 -minor-free graphs, which is due to Lovasz, Seymour, Schrijver, and Truemper.