An even-cycle decomposition   of a graph GG is a partition of E(G)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 K5K5-minor.#R##N##R##N#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-K4K4-minor-free graph with an even number of edges has an even-cycle decomposition.#R##N##R##N#This is best possible in the sense that ‘odd-K4K4-minor-free’ cannot be replaced with ‘odd-K5K5-minor-free.’ The main technical ingredient is a structural characterization of the class of odd-K4K4-minor-free graphs, which is due to Lovasz, Seymour, Schrijver, and Truemper.
['Tony Huynh', 'Sang-il Oum', 'Maryam Verdian-Rizi']
Even-cycle decompositions of graphs with no odd-$K_4$-minor