Even-cycle decompositions of graphs with no odd--minor
arXiv:1211.1868 · doi:10.1016/j.ejc.2017.04.010
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 -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--minor-free graph with an even number of edges has an even-cycle decomposition. This is best possible in the sense that `odd--minor-free' cannot be replaced with `odd--minor-free.' The main technical ingredient is a structural characterization of the class of odd--minor-free graphs, which is due to Lovász, Seymour, Schrijver, and Truemper.
17 pages, 6 figures; minor revision