  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 Lov\'asz, Seymour, Schrijver, and Truemper.
