def edge_coloring(self):
r"""
Compute a proper edge-coloring.
A proper edge-coloring is an assignment of colors to the sets of the
hypergraph such that two sets with non-empty intersection receive
different colors. The coloring returned minimizes the number of colors.
OUTPUT:
A partition of the sets into color classes.
EXAMPLES::
sage: H = Hypergraph([{1,2,3},{2,3,4},{3,4,5},{4,5,6}]); H
Hypergraph on 6 vertices containing 4 sets
sage: C = H.edge_coloring()
sage: C # random
[[{3, 4, 5}], [{4, 5, 6}, {1, 2, 3}], [{2, 3, 4}]]
sage: Set(sum(C,[])) == Set(H._sets)
True
"""
from sage.graphs.graph import Graph
g = Graph([self._sets,lambda x,y : len(x&y)],loops = False)
return g.coloring(algorithm="MILP")
def edge_coloring(self):
r"""
Compute a proper edge-coloring.
A proper edge-coloring is an assignment of colors to the sets of the
incidence structure such that two sets with non-empty intersection
receive different colors. The coloring returned minimizes the number of
colors.
OUTPUT:
A partition of the sets into color classes.
EXAMPLES::
sage: H = Hypergraph([{1,2,3},{2,3,4},{3,4,5},{4,5,6}]); H
Incidence structure with 6 points and 4 blocks
sage: C = H.edge_coloring()
sage: C # random
[[[3, 4, 5]], [[2, 3, 4]], [[4, 5, 6], [1, 2, 3]]]
sage: Set(map(Set,sum(C,[]))) == Set(map(Set,H.blocks()))
True
"""
from sage.graphs.graph import Graph
blocks = self.blocks()
blocks_sets = map(frozenset,blocks)
g = Graph([range(self.num_blocks()),lambda x,y : len(blocks_sets[x]&blocks_sets[y])],loops = False)
return [[blocks[i] for i in C] for C in g.coloring(algorithm="MILP")]
def edge_coloring(self):
r"""
Compute a proper edge-coloring.
A proper edge-coloring is an assignment of colors to the sets of the
hypergraph such that two sets with non-empty intersection receive
different colors. The coloring returned minimizes the number of colors.
OUTPUT:
A partition of the sets into color classes.
EXAMPLES::
sage: H = Hypergraph([{1,2,3},{2,3,4},{3,4,5},{4,5,6}]); H
Hypergraph on 6 vertices containing 4 sets
sage: C = H.edge_coloring()
sage: C # random
[[{3, 4, 5}], [{4, 5, 6}, {1, 2, 3}], [{2, 3, 4}]]
sage: Set(sum(C,[])) == Set(H._sets)
True
"""
from sage.graphs.graph import Graph
g = Graph([self._sets, lambda x, y: len(x & y)], loops=False)
return g.coloring(algorithm="MILP")
def edge_coloring(self):
r"""
Compute a proper edge-coloring.
A proper edge-coloring is an assignment of colors to the sets of the
incidence structure such that two sets with non-empty intersection
receive different colors. The coloring returned minimizes the number of
colors.
OUTPUT:
A partition of the sets into color classes.
EXAMPLES::
sage: H = Hypergraph([{1,2,3},{2,3,4},{3,4,5},{4,5,6}]); H
Incidence structure with 6 points and 4 blocks
sage: C = H.edge_coloring()
sage: C # random
[[[3, 4, 5]], [[2, 3, 4]], [[4, 5, 6], [1, 2, 3]]]
sage: Set(map(Set,sum(C,[]))) == Set(map(Set,H.blocks()))
True
"""
from sage.graphs.graph import Graph
blocks = self.blocks()
blocks_sets = map(frozenset, blocks)
g = Graph([
range(self.num_blocks()),
lambda x, y: len(blocks_sets[x] & blocks_sets[y])
],
loops=False)
return [[blocks[i] for i in C] for C in g.coloring(algorithm="MILP")]
