Can we have a poly time reduction from 2-SAT to 2-Coloring problem?

I know that given a 2-Coloring instance we can easily convert it into a 2-Sat instance in polynomial time . Is the reverse possible? i.e given a 2-sat instance can we convert it into an 2-Coloring instance in poly time?

As rotia mentions in their comment, to make the question meaningful we must restrict the power of the allowable reductions. The two most obvious choices are to allow only logspace reductions or only AC0 reductions.

Since 2SAT is NL-complete (with respect to logspace reductions) whereas 2COL (2-colorability) is in L (this is a non-trivial result which follows from Reingold's theorem), if there is a logspace reduction from 2SAT to 2COL then L=NL, which is considered unlikely. Hence we do not expect such a reduction to exist; a fortiori we do not expect an AC0 reduction to exist.

StackExchange Computer Science Q#63973, answer score: 9