I want to show that $ USTCON\le_L CO-2Col$ (Log-Space reduction)

$ USTCON$ The $ s-t$ connectivity problem for **undirected** graphs is called $ USTCON$ .

Input: An undirected graph $ G=(V,E)$ , $ s,t \in V$ . Output: 1 iff $ s$ is connected to $ t$ in $ G$ .

$ CO-2Col$ A graph is $ 2$ -colorable if there is a way to color the vertices of $ G$ with $ 2$ colors, such that for every edge the two vertices on the edge are colored differently. $ CO-2Col$ is the following problem:

Input: An undirected graph $ G$ . Output: 1 iff $ G$ is NOT $ 2$ -colorable.

I know this can be solved by simply using the $ USTCON$ decision algorithm and then output a bipartite or non-bipartite graph accordingly. But is there any other reduction that actually changes the input graph?