Log-Space Reduction $CO-2Col \le_L USTCON$
Hint: Extend the input graph to a connected graph which is bipartite iff the original graph is bipartite.

If the original graph is not bipartite, then the new graph will a fortiori not be bipartite. The more difficult direction is ensuring that when the original is bipartite, the new edges do not render it non-bipartite. This property is satisfied as long as the edges that you introduce do not close an odd cycle.

Further hint: Take $n$ copies of your original graph (where $n$ is the number of vertices), and add a new vertex $s$. Connect $s$ to the $i$th vertex in the $i$th copy.