I was reading through a textbook that included the following problem but did not include an answer, probably to sell the answer in a separate booklet.
An n-coloring $C$ of an undirected graph $G(V, E)$ is an assignment of colors to each node, where the color is drawn from ${c_1, c_2, . . . , c_n}$. Define improper edges as those with both end-points having the same color. Let $I(C)$ be the number of improper edges in $C$.
Design a Markov chain using the Metropolis algorithm in such a way that in the stationary distribution, the probability of coloring $C$ is proportional to $λ^{I(C)}$ for given $λ > 0$. Pairs of states of the chain are connected if the corresponding colorings differ in that of just one vertex.
I did my best trying to solve the question, but apparently my best was not enough. I would appreciate it if a smarter head than mine can explain the answer to me.