def statistic(G, fast=True):
    """

    Proposition 2.3 of Lovász, László, J. Nešetšil, and Ales
    Pultr. "On a product dimension of graphs." Journal of
    Combinatorial Theory, Series B 29.1 (1980): 47-67.::

        sage: N = 7; l = [G for n in range(1, N) for G in graphs(n) if G.complement().chromatic_index() <= 1]
        sage: all(statistic(G) == G.complement().chromatic_index() + 1 for G in l)
        True
        sage: N = 6; l = [G for n in range(1, N) for G in graphs(n) if G.complement().chromatic_index() > 1]
        sage: all(statistic(G) <= G.complement().chromatic_index() for G in l)
        True
        sage: all(statistic(G) == G.complement().chromatic_index() for G in l if G.complement().is_triangle_free())
        True

    Proposition 3.6::

        sage: N = 8; l = [(k, n, graphs.CompleteGraph(n) + Graph(k)) for k in range(1, N) for n in range(2, N)]
        sage: all(statistic(G) == (n+1 if k > factorial(n-1) else n) for k, n, G in l)
        True

    TESTS::

        sage: N = 6; all(statistic(G) == statistic(G, False) for n in range(N) for G in graphs(n))
        True
    """
    if fast:
        Gc = G.complement()
        Gc_chi = Gc.chromatic_index()
        if Gc_chi <= 1:
            return Gc_chi + 1
        if Gc.is_triangle_free():
            return Gc_chi

        lG = sorted(G.connected_components_subgraphs(), key=lambda G: G.num_verts())
        if len(lG) > 1 and lG[-2].num_verts() == 1 and lG[-1].is_clique():
            if len(lG) - 1 <= factorial(lG[-1].num_verts()-1):
                return lG[-1].num_verts()
            return lG[-1].num_verts() + 1

    d = 0
    n = G.num_verts()
    K = graphs.CompleteGraph(n)
    H = K
    while True:
        d += 1
        if H.subgraph_search(G, induced=True) is not None:
            return d
        H = H.categorical_product(K)