Finding Even Cycles Even Faster

We describe efficient algorithms for finding even cycles in undirected graphs. Our main results are the following: (i) For every $k \geq 2$, there is an $O(V^2)$ time algorithm that decides whether an undirected graph $G=(V,E)$ contains a simple cycle of length $2k$, and finds one if it does. (ii) There is an $O(V^2)$ time algorithm that finds a shortest even cycle in an undirected graph $G=(V,E)$.