Is there a relationship between the number of connected components in a bigraph and the number of possible 2-colorings?
A connected bigraph (i.e. only one component) can be 2-colored in exactly two different ways. It seems like a bigraph with two components can be 2-colored in four different ways; and three components, eight ways.
This suggests that a bigragph with $n$ components can be 2-colored in $2^n$ ways. Is that correct? Does this mean a k-partite graph with $n$ components can be k-colored in $k^n$ ways?
So you mean a bipartite graph ?
If so, that sounds correct ! No shame in asking.
Though It's only true for $k = 2$. In general in a $k$-partite graph, one component can have a lot more than $k$ possible colorings.
Take a triangle and add a vertex $v$ with one single neighbor on this triangle. It's a 3-colorable component, but for each of the 3 proper 3-colorings of the triangle, $v$ has 2 available colors. So in total this graph gets 6 colorings.