Parameterized complexity of even/odd subgraph problems

We study the parameterized complexity of the problems of determining whether a graph contains a k-edge subgraph (k-vertex induced subgraph) that is a Π-graph for Π-graphs being one of the following four classes of graphs: Eulerian graphs, even graphs, odd graphs, and connected odd graphs. We also consider the parameterized complexity of their parametric dual problems.For these sixteen problems, we show that eight of them are fixed parameter tractable and four are W[1]-hard. Our main techniques are the color-coding method of Alon, Yuster and Zwick, and the random separation method of Cai, Chan and Chan.