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

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 K5-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-K4-minor-free graph with an even number of edges has an even-cycle decomposition. This is best possible in the sense that ‘odd-K4-minor-free’ cannot be replaced with ‘odd-K5-minor-free.’ The main technical ingredient is a structural characterization of the class of odd-K4-minor-free graphs, which is due to Lovász, Seymour, Schrijver, and Truemper.