Identifier
Identifier
Values
([(0,2),(1,2)],3) generating graphics... => 2
([(0,1),(0,2),(1,2)],3) generating graphics... => 3
([(1,3),(2,3)],4) generating graphics... => 2
([(1,2),(1,3),(2,3)],4) generating graphics... => 3
([(0,3),(1,3),(2,3)],4) generating graphics... => 3
([(0,3),(1,2),(2,3)],4) generating graphics... => 2
([(0,3),(1,2),(1,3),(2,3)],4) generating graphics... => 2
([(0,2),(0,3),(1,2),(1,3)],4) generating graphics... => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) generating graphics... => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) generating graphics... => 3
([(1,4),(2,4),(3,4)],5) generating graphics... => 3
([(0,4),(1,4),(2,4),(3,4)],5) generating graphics... => 4
([(1,4),(2,3),(3,4)],5) generating graphics... => 2
([(1,4),(2,3),(2,4),(3,4)],5) generating graphics... => 2
([(1,3),(1,4),(2,3),(2,4)],5) generating graphics... => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) generating graphics... => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) generating graphics... => 3
([(0,4),(1,4),(2,3),(3,4)],5) generating graphics... => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) generating graphics... => 2
([(0,4),(1,3),(2,3),(2,4)],5) generating graphics... => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) generating graphics... => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) generating graphics... => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) generating graphics... => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) generating graphics... => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) generating graphics... => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) generating graphics... => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) generating graphics... => 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) generating graphics... => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) generating graphics... => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) generating graphics... => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) generating graphics... => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) generating graphics... => 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) generating graphics... => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) generating graphics... => 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) generating graphics... => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) generating graphics... => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) generating graphics... => 2
([(1,5),(2,5),(3,5),(4,5)],6) generating graphics... => 4
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) generating graphics... => 5
([(1,5),(2,5),(3,4),(4,5)],6) generating graphics... => 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(1,5),(2,4),(3,4),(3,5)],6) generating graphics... => 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) generating graphics... => 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) generating graphics... => 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) generating graphics... => 3
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) generating graphics... => 2
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) generating graphics... => 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) generating graphics... => 2
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) generating graphics... => 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) generating graphics... => 3
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 3
([(0,5),(1,5),(2,4),(3,4)],6) generating graphics... => 3
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) generating graphics... => 3
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) generating graphics... => 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) generating graphics... => 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) generating graphics... => 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) generating graphics... => 3
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 3
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) generating graphics... => 3
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) generating graphics... => 2
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) generating graphics... => 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6) generating graphics... => 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) generating graphics... => 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) generating graphics... => 2
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) generating graphics... => 2
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) generating graphics... => 4
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6) generating graphics... => 2
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) generating graphics... => 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) generating graphics... => 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) generating graphics... => 2
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) generating graphics... => 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) generating graphics... => 2
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6) generating graphics... => 2
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) generating graphics... => 2
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) generating graphics... => 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,5),(1,4),(2,3),(2,4),(3,5)],6) generating graphics... => 2
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 1
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) generating graphics... => 2
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) generating graphics... => 2
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) generating graphics... => 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) generating graphics... => 2
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6) generating graphics... => 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) generating graphics... => 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) generating graphics... => 2
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) generating graphics... => 2
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) generating graphics... => 2
([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) generating graphics... => 2
([(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) generating graphics... => 2
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) generating graphics... => 2
([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6) generating graphics... => 2
([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) generating graphics... => 2
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6) generating graphics... => 2
([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6) generating graphics... => 2
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) generating graphics... => 2
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6) generating graphics... => 2
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6) generating graphics... => 2
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) generating graphics... => 2
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6) generating graphics... => 2
([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) generating graphics... => 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) generating graphics... => 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) generating graphics... => 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) generating graphics... => 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) generating graphics... => 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) generating graphics... => 2
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
References
[1] Kalinowski, Rafał, Pilśniak, M. Distinguishing graphs by edge-colourings MathSciNet:3286626
Code
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]
    """
    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
Created
Dec 12, 2017 at 11:00 by Martin Rubey
Updated
Dec 12, 2017 at 11:00 by Martin Rubey