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Solve the following problem using Python, implementing the functions described below, one line at a time
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Description:
 def _partial_precolor(G, chi_ub):
 """In order to reduce the number of variables in the QUBO, we want to color as many nodes as possible without affecting the min vertex coloring. Without loss of generality, we can choose a single maximal clique and color each node in it uniquely. Returns ------- partial_coloring : dict A dict describing a partial coloring of the nodes of G. Of the form possible_colors : dict A dict giving the possible colors for each node in G not already chi_lb : int A lower bound on the chromatic number chi. Notes ----- partial_coloring.keys() and possible_colors.keys() should be disjoint. """


    # find a random maximal clique and give each node in it a unique color
    v = next(iter(G))
    clique = [v]
    for u in G[v]:
        if all(w in G[u] for w in clique):
            clique.append(u)

    partial_coloring = {v: c for c, v in enumerate(clique)}
    chi_lb = len(partial_coloring)  # lower bound for the chromatic number

    # now for each uncolored node determine the possible colors
    possible_colors = {v: set(range(chi_ub)) for v in G if v not in partial_coloring}

    for v, color in iteritems(partial_coloring):
        for u in G[v]:
            if u in possible_colors:
                possible_colors[u].discard(color)

    # TODO: there is more here that can be done. For instance some nodes now
    # might only have one possible color. Or there might only be one node
    # remaining to color

    return partial_coloring, possible_colors, chi_lb