How may an algorithm always color optimally connected bipartite graphs?
You can prove an even stronger claim. Consider the following algorithm:

Let $v_1,\ldots,v_n$ be an ordering of the vertices such that for $i&gt;1$, each $v_i$ is connected to at least one of $v_1,\ldots,v_{i-1}$.
Go over all vertices $v_1,\ldots,v_n$ in order, and color each one with the smallest available color.

This algorithm also uses (at most) two colors to color a connected bipartite graph. The property of the ordering that I highlight, which is satisfied by the ordering produced by the Dsatur algorithm, suffices to guarantee that the algorithm uses the minimal number of colors.
To prove that this algorithm works for a connected bipartite graph with bipartition $V_1 \cup V_2$, where $v_1 \in V_1$, prove by induction that if $v_i \in V_j$ then $v_i$ gets colored $j$.