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Statistic identifier: St001642

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Collection: Graphs

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Description: The Prague dimension of a graph.

This is the least number of complete graphs such that the graph is an induced subgraph of their (categorical) product.

Put differently, this is the least number $n$ such that the graph can be embedded into $\mathbb N^n$, where two points are connected by an edge if and only if they differ in all coordinates.

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References: [1]   Lovász, L., Nešetřil, J., Pultr, A. On a product dimension of graphs [[MathSciNet:0584160]]

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Code:
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)


-----------------------------------------------------------------------------
Statistic values:

([],1)                                                                                          => 1
([],2)                                                                                          => 2
([(0,1)],2)                                                                                     => 1
([],3)                                                                                          => 2
([(1,2)],3)                                                                                     => 2
([(0,2),(1,2)],3)                                                                               => 2
([(0,1),(0,2),(1,2)],3)                                                                         => 1
([],4)                                                                                          => 2
([(2,3)],4)                                                                                     => 3
([(1,3),(2,3)],4)                                                                               => 2
([(0,3),(1,3),(2,3)],4)                                                                         => 2
([(0,3),(1,2)],4)                                                                               => 2
([(0,3),(1,2),(2,3)],4)                                                                         => 2
([(1,2),(1,3),(2,3)],4)                                                                         => 3
([(0,3),(1,2),(1,3),(2,3)],4)                                                                   => 2
([(0,2),(0,3),(1,2),(1,3)],4)                                                                   => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)                                                             => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)                                                       => 1
([],5)                                                                                          => 2
([(3,4)],5)                                                                                     => 3
([(2,4),(3,4)],5)                                                                               => 3
([(1,4),(2,4),(3,4)],5)                                                                         => 2
([(0,4),(1,4),(2,4),(3,4)],5)                                                                   => 2
([(1,4),(2,3)],5)                                                                               => 3
([(1,4),(2,3),(3,4)],5)                                                                         => 3
([(0,1),(2,4),(3,4)],5)                                                                         => 3
([(2,3),(2,4),(3,4)],5)                                                                         => 3
([(0,4),(1,4),(2,3),(3,4)],5)                                                                   => 3
([(1,4),(2,3),(2,4),(3,4)],5)                                                                   => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)                                                             => 3
([(1,3),(1,4),(2,3),(2,4)],5)                                                                   => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)                                                             => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)                                                             => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)                                                             => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)                                                       => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)                                                       => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)                                                 => 2
([(0,4),(1,3),(2,3),(2,4)],5)                                                                   => 2
([(0,1),(2,3),(2,4),(3,4)],5)                                                                   => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)                                                             => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)                                                       => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)                                                             => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)                                                       => 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)                                                 => 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)                                                       => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)                                                       => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)                                                 => 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)                                           => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)                                                 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)                                           => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)                                     => 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)                               => 1
([],6)                                                                                          => 2
([(4,5)],6)                                                                                     => 3
([(3,5),(4,5)],6)                                                                               => 3
([(2,5),(3,5),(4,5)],6)                                                                         => 3
([(1,5),(2,5),(3,5),(4,5)],6)                                                                   => 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)                                                             => 2
([(2,5),(3,4)],6)                                                                               => 3
([(2,5),(3,4),(4,5)],6)                                                                         => 3
([(1,2),(3,5),(4,5)],6)                                                                         => 3
([(3,4),(3,5),(4,5)],6)                                                                         => 4
([(1,5),(2,5),(3,4),(4,5)],6)                                                                   => 3
([(0,1),(2,5),(3,5),(4,5)],6)                                                                   => 3
([(2,5),(3,4),(3,5),(4,5)],6)                                                                   => 3
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)                                                             => 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)                                                             => 3
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)                                                       => 3
([(2,4),(2,5),(3,4),(3,5)],6)                                                                   => 3
([(0,5),(1,5),(2,4),(3,4)],6)                                                                   => 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)                                                             => 3
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)                                                             => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                                             => 3
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)                                                             => 3
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)                                                             => 3
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)                                                       => 3
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                                       => 3
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)                                                       => 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                                 => 3
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)                                                       => 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)                                                       => 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)                                                 => 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                                 => 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                                 => 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                           => 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)                                           => 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                     => 2
([(0,5),(1,4),(2,3)],6)                                                                         => 3
([(1,5),(2,4),(3,4),(3,5)],6)                                                                   => 3
([(0,1),(2,5),(3,4),(4,5)],6)                                                                   => 3
([(1,2),(3,4),(3,5),(4,5)],6)                                                                   => 3
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)                                                             => 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)                                                             => 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)                                                             => 3
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)                                                       => 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)                                                       => 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)                                                 => 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)                                                             => 3
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)                                                       => 3
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)                                                       => 3
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)                                                       => 3
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)                                                       => 3
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)                                                       => 3
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)                                                 => 3
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                                 => 3
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)                                                 => 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)                                           => 3
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)                                                             => 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)                                                             => 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)                                                             => 4
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)                                                       => 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)                                                       => 3
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)                                                       => 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                                       => 3
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)                                                       => 3
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)                                                 => 3
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)                                                 => 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)                                                 => 3
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                           => 3
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                                       => 4
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)                                                 => 3
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                                 => 4
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                           => 3
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)                                                 => 3
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)                                           => 3
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                           => 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)                                           => 3
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                           => 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)                                     => 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)                                           => 3
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                           => 4
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                           => 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                     => 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)                                     => 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                               => 3
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)                                                 => 3
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)                                                 => 3
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)                                           => 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)                                           => 3
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)                                     => 2
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)                                                       => 2
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)                                                 => 2
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)                                                 => 2
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)                                                 => 3
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)                                                 => 3
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)                                                 => 3
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                           => 3
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)                                           => 3
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)                                           => 2
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)                                     => 2
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)                                           => 2
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)                                           => 3
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)                                     => 2
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                     => 3
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                               => 2
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                     => 3
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                     => 4
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                     => 3
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                               => 3
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)                                     => 2
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)                                     => 2
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                               => 2
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                         => 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)                                     => 2
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                               => 2
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)                               => 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)                         => 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                   => 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)                                                       => 3
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)                                                 => 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)                                                 => 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                                 => 4
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)                                           => 3
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                           => 4
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                     => 3
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)                                           => 3
([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)                                     => 3
([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                     => 4
([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                     => 3
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                                     => 3
([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                               => 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)                               => 2
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                         => 2
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6)                                           => 3
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)                                           => 3
([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)                                     => 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)                               => 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)                                     => 2
([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                               => 2
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)                               => 3
([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                         => 2
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                   => 2
([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)                               => 3
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                               => 3
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                         => 3
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                               => 4
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                               => 5
([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                         => 4
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                   => 3
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)             => 2
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)                         => 2
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)                         => 3
([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)                         => 2
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)                   => 2
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)                   => 2
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)             => 2
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)       => 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1

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Created: Nov 19, 2020 at 10:35 by Martin Rubey

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Last Updated: Nov 19, 2020 at 15:52 by Martin Rubey