"""
Ramsey numbers.
"""
import networkx as nx
from ...utils import arbitrary_element
__all__ = ["ramsey_R2"]
[docs]def ramsey_R2(G):
r"""Compute the largest clique and largest independent set in `G`.
This can be used to estimate bounds for the 2-color
Ramsey number `R(2;s,t)` for `G`.
This is a recursive implementation which could run into trouble
for large recursions. Note that self-loop edges are ignored.
Parameters
----------
G : NetworkX graph
Undirected graph
Returns
-------
max_pair : (set, set) tuple
Maximum clique, Maximum independent set.
"""
if not G:
return set(), set()
node = arbitrary_element(G)
nbrs = (nbr for nbr in nx.all_neighbors(G, node) if nbr != node)
nnbrs = nx.non_neighbors(G, node)
c_1, i_1 = ramsey_R2(G.subgraph(nbrs).copy())
c_2, i_2 = ramsey_R2(G.subgraph(nnbrs).copy())
c_1.add(node)
i_2.add(node)
# Choose the larger of the two cliques and the larger of the two
# independent sets, according to cardinality.
return max(c_1, c_2, key=len), max(i_1, i_2, key=len)