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Statistic identifier: St001703
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Collection: Graphs
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Description: The villainy of a graph.
The villainy of a permutation of a proper coloring $c$ of a graph is the minimal Hamming distance between $c$ and a proper coloring.
The villainy of a graph is the maximal villainy of a permutation of a proper coloring.
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References: [1] Jahanbekam, S., Lin, M.-R. Characterization of Graphs with Villainy 2 [[arXiv:2103.05816]]
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Code:
# extremely stupid code
def is_proper_coloring(G, c):
"""
assume that G has vertices {0,...,n-1}.
"""
return not any(c[v] in [c[u] for u in G[v]] for v in G)
def villainy_coloring(G, c):
"""
assume that G has vertices {0,...,n-1}.
"""
villainy = None
for c_pi in Permutations(c):
if is_proper_coloring(G, c_pi):
d = sum(1 for c1, c2 in zip(c, c_pi) if c1 != c2)
if villainy is None or d < villainy:
villainy = d
if villainy == 0:
return villainy
return villainy
def statistic(G):
G = G.relabel(inplace=False)
n = G.num_verts()
chi = G.chromatic_number()
villainy = 0
for c in sage.graphs.graph_coloring.all_graph_colorings(G, chi, vertex_color_dict=True):
c = [c[i] for i in range(n)]
for c_pi in Permutations(c):
d = villainy_coloring(G, c_pi)
villainy = max(villainy, d)
return villainy
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Statistic values:
([],0) => 0
([],1) => 0
([],2) => 0
([(0,1)],2) => 0
([],3) => 0
([(1,2)],3) => 2
([(0,2),(1,2)],3) => 2
([(0,1),(0,2),(1,2)],3) => 0
([],4) => 0
([(2,3)],4) => 2
([(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) => 2
([(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) => 4
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0
([],5) => 0
([(3,4)],5) => 2
([(2,4),(3,4)],5) => 2
([(1,4),(2,4),(3,4)],5) => 4
([(0,4),(1,4),(2,4),(3,4)],5) => 2
([(1,4),(2,3)],5) => 2
([(1,4),(2,3),(3,4)],5) => 2
([(0,1),(2,4),(3,4)],5) => 4
([(2,3),(2,4),(3,4)],5) => 4
([(0,4),(1,4),(2,3),(3,4)],5) => 4
([(1,4),(2,3),(2,4),(3,4)],5) => 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 4
([(1,3),(1,4),(2,3),(2,4)],5) => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
([(0,4),(1,3),(2,3),(2,4)],5) => 4
([(0,1),(2,3),(2,4),(3,4)],5) => 2
([(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) => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 3
([(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) => 2
([(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),(3,4)],5) => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
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Created: Mar 11, 2021 at 13:16 by Martin Rubey
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Last Updated: Mar 11, 2021 at 13:16 by Martin Rubey