This is the complete bipartite graph K 2,4 . K 1 does not have enough vertices to be bipartite. Clearly, K 2 is not bipartite. If n is even, then C n is bipartite, since we can take one partition to be every other vertex. If n is odd, then C n is not bipartite.
Every wheel contains triangles, so no W n is bipartite.
The 4-wheel with one edge along the rim deleted is such a graph.