This Thesis deals with discrete Markov chains and their applications in the generation of combinatorial designs. A conjecture on the generation of proper edge colorings of the complete graph K_n, for n even, is tackled. A proper edge coloring is an edge coloring such that no two adjacent edges have the same color. Proper edge colorings are characterized by minimizing the potential and maximizing the entropy. We implement an algorithm in the software R to generate proper colorings from any arbitrary coloring of K_n, by identifying the colorings as nodes in a Markov chain, where transition probabilities are defined so that the potential decreases or, alternatively, the entropy increases. The conjecture states that the algorithm converges in polynomial time. We give original proofs of the conjecture in K_4 and K_6, and we provide new results and ideas that could be used to prove the conjecture in the general case K_n.