In this paper, we exhibit an irreducible Markov chain $M$ on the edge $k$-colorings of bipartite graphs based on certain properties of the solution space. We show that diameter of this Markov chain grows linearly with the number of edges in the graph. We also prove a polynomial upper bound on the inverse of acceptance ratio of the Metropolis-Hastings algorithm when the algorithm is applied on $M$ with the uniform distribution of all possible edge $k$-colorings of $G$. A special case of our results is the solution space of the possible completions of Latin rectangles.