Values
=>
Cc0020;cc-rep
([],1)=>0
([],2)=>0
([(0,1)],2)=>1
([],3)=>0
([(1,2)],3)=>1
([(0,2),(1,2)],3)=>1
([(0,1),(0,2),(1,2)],3)=>2
([],4)=>0
([(2,3)],4)=>1
([(1,3),(2,3)],4)=>1
([(0,3),(1,3),(2,3)],4)=>1
([(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)=>1
([(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)=>3
([],5)=>0
([(3,4)],5)=>1
([(2,4),(3,4)],5)=>1
([(1,4),(2,4),(3,4)],5)=>1
([(0,4),(1,4),(2,4),(3,4)],5)=>1
([(1,4),(2,3)],5)=>2
([(1,4),(2,3),(3,4)],5)=>2
([(0,1),(2,4),(3,4)],5)=>2
([(2,3),(2,4),(3,4)],5)=>2
([(0,4),(1,4),(2,3),(3,4)],5)=>2
([(1,4),(2,3),(2,4),(3,4)],5)=>2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)=>2
([(1,3),(1,4),(2,3),(2,4)],5)=>1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)=>2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>2
([(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)=>1
([(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)=>3
([(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)=>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)=>3
([(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)=>3
([(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)=>3
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>4
([],6)=>0
([(4,5)],6)=>1
([(3,5),(4,5)],6)=>1
([(2,5),(3,5),(4,5)],6)=>1
([(1,5),(2,5),(3,5),(4,5)],6)=>1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1
([(2,5),(3,4)],6)=>2
([(2,5),(3,4),(4,5)],6)=>2
([(1,2),(3,5),(4,5)],6)=>2
([(3,4),(3,5),(4,5)],6)=>2
([(1,5),(2,5),(3,4),(4,5)],6)=>2
([(0,1),(2,5),(3,5),(4,5)],6)=>2
([(2,5),(3,4),(3,5),(4,5)],6)=>2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>2
([(2,4),(2,5),(3,4),(3,5)],6)=>1
([(0,5),(1,5),(2,4),(3,4)],6)=>2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)=>2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)=>2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)=>2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)=>2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>2
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>1
([(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)=>2
([(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)=>1
([(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)=>2
([(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)=>2
([(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)=>2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)=>3
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)=>2
([(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)=>3
([(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)=>3
([(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)=>2
([(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)=>2
([(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)=>3
([(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)=>2
([(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)=>2
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>2
([(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)=>2
([(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)=>3
([(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
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>3
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)=>3
([(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)=>3
([(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),(1,3),(2,3),(4,5)],6)=>4
([(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)=>4
([(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)=>4
([(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)=>4
([(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
([(0,1),(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)=>4
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>4
([(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)=>4
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>3
([(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),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
([(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)=>4
([(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)=>4
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>3
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Description
The biclique partition number of a graph.
The biclique partition number of a graph is the minimum number of pairwise edge disjoint complete bipartite subgraphs so that each edge belongs to exactly one of them. A theorem of Graham and Pollak [1] asserts that the complete graph $K_n$ has biclique partition number $n - 1$.
The biclique partition number of a graph is the minimum number of pairwise edge disjoint complete bipartite subgraphs so that each edge belongs to exactly one of them. A theorem of Graham and Pollak [1] asserts that the complete graph $K_n$ has biclique partition number $n - 1$.
References
[1] MR0289210
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[2] WARNING - could not verify link HTTP Error 404: Not Found
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Code
@cached_function
def statistic(G, certificate=False, verbose=False):
"""
sage: [statistic(graphs.CompleteGraph(n)) for n in range(1,6)]
[0, 1, 2, 3, 4]
sage: [statistic(graphs.PathGraph(n)) for n in range(1,8)]
[0, 1, 1, 2, 2, 3, 3]
sage: [statistic(graphs.CycleGraph(n)) for n in range(1,8)]
[0, 1, 2, 1, 3, 3, 4]
sage: n=5; all(statistic(G) <= n-G.independent_set(value_only=True) for G in graphs(n))
True
"""
E = G.edges(labels=False)
if not E:
return 0
for c in range(1, len(E)+1):
for p in SetPartitions(E, c):
if verbose: print("P", p)
for b in p:
V = [u for e in b for u in e]
H = G.subgraph(edges=b, vertices=V)
if verbose: print("V", H.vertices(), "E", H.edges(labels=False))
is_bipartite, bipartition = H.is_bipartite(certificate=True)
if is_bipartite:
if verbose: print(bipartition)
k = list(bipartition.values()).count(0)
l = H.num_verts() - k
B = graphs.CompleteBipartiteGraph(k, l)
if not H.is_isomorphic(B):
if verbose: print("not complete")
break
else:
if verbose: print("not bipartite")
break
else:
if certificate:
return c, p
return c
Created
Jun 28, 2022 at 12:09 by Martin Rubey
Updated
Jun 28, 2022 at 12:09 by Martin Rubey
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