Bipartite graph and cycle of even length

Question

A bipartite graph is a graph whose vertices can be divided into two disjoint sets U and V such that every edge connects a vertex in U to one in V; that is, U and V are each independent sets. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles.

Can I also say that if in a graph G all cycles are even in length, then it is bipartite?

I thought of one graph of even length cycle and it turned out to be non-bipartite.

     1----------2
     |          |
     |          |
     |          |
     |          |
     3----------4

Answer 1

If in a graph G all cycles are even in length, then it is bipartite.

Apply BFS algorithm to graph G. It divides vertices of G into layers. Set U consists of vertices from odd layers, V of vertices from even layers. Let's assume (by contradiction) that there exists edge e that connects some two vertices x, y from U. Let r be the root of tree determined by BFS algorithm. Then path from x to r, from r to y, and edge e are cycle of odd length - that's contradiction as graph G doesn't contain odd-length cycles. (same with set V).