def statistic(G):
    """the smallest number of colours such that there is a colouring of
    the edges which is not preserved by any automorphism.

    sage: [statistic(graphs.CycleGraph(r)) for r in range(3,7)]
    [3, 3, 3, 2]

    sage: [statistic(graphs.CompleteGraph(r)) for r in range(3,8)]
    [3, 3, 3, 2, 2]

    sage: [statistic(graphs.CompleteBipartiteGraph(1,r)) for r in range(3,8)]
    [3, 4, 5, 6, 7]
    """
    G = G.copy(immutable=False)
    for c in range(G.num_edges()+1):
        # try to find a colouring
        V = G.edges(labels=False)
        # partition the set of edges into c parts
        for colouring in SetPartitions(V, c):
            for i, block in enumerate(colouring):
                for u, v in block:
                    G.set_edge_label(u,v,i)
            if G.automorphism_group(edge_labels=True, order=True)[1] == 1:
                return c