***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * This information is distributed in the hope that it will be useful, * * but WITHOUT ANY WARRANTY; without even the implied warranty of * * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. * ***************************************************************************** ----------------------------------------------------------------------------- Statistic identifier: St001812 ----------------------------------------------------------------------------- Collection: Graphs ----------------------------------------------------------------------------- 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 $n - 1$. ----------------------------------------------------------------------------- References: [1] [[MR0289210]] WARNING - could not verify link HTTP Error 404: Not Found [2] WARNING - could not verify link HTTP Error 404: Not Found ----------------------------------------------------------------------------- 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 ----------------------------------------------------------------------------- Statistic values: ([],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 ----------------------------------------------------------------------------- Created: Jun 28, 2022 at 12:09 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Jun 28, 2022 at 12:09 by Martin Rubey