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?