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Identifier

St001670: Graphs ⟶ ℤ

Values

([],1) => 1

([],2) => 1

([(0,1)],2) => 2

([],3) => 1

([(1,2)],3) => 2

([(0,2),(1,2)],3) => 2

([(0,1),(0,2),(1,2)],3) => 3

([],4) => 1

([(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) => 3

([(1,2),(1,3),(2,3)],4) => 3

([(0,3),(1,2),(1,3),(2,3)],4) => 3

([(0,2),(0,3),(1,2),(1,3)],4) => 3

([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3

([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4

([],5) => 1

([(3,4)],5) => 2

([(2,4),(3,4)],5) => 2

([(1,4),(2,4),(3,4)],5) => 2

([(0,4),(1,4),(2,4),(3,4)],5) => 2

([(1,4),(2,3)],5) => 2

([(1,4),(2,3),(3,4)],5) => 3

([(0,1),(2,4),(3,4)],5) => 3

([(2,3),(2,4),(3,4)],5) => 3

([(0,4),(1,4),(2,3),(3,4)],5) => 3

([(1,4),(2,3),(2,4),(3,4)],5) => 3

([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 3

([(1,3),(1,4),(2,3),(2,4)],5) => 3

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 3

([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3

([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 3

([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 3

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3

([(0,4),(1,3),(2,3),(2,4)],5) => 3

([(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) => 3

([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 4

([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 4

([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4

([(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),(3,4)],5) => 4

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 4

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 4

([(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) => 5

([],6) => 1

([(4,5)],6) => 2

([(3,5),(4,5)],6) => 2

([(2,5),(3,5),(4,5)],6) => 2

([(1,5),(2,5),(3,5),(4,5)],6) => 2

([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2

([(2,5),(3,4)],6) => 2

([(2,5),(3,4),(4,5)],6) => 3

([(1,2),(3,5),(4,5)],6) => 3

([(3,4),(3,5),(4,5)],6) => 3

([(1,5),(2,5),(3,4),(4,5)],6) => 3

([(0,1),(2,5),(3,5),(4,5)],6) => 3

([(2,5),(3,4),(3,5),(4,5)],6) => 3

([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 3

([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 3

([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 3

([(2,4),(2,5),(3,4),(3,5)],6) => 3

([(0,5),(1,5),(2,4),(3,4)],6) => 3

([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 3

([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 3

([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3

([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 3

([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 3

([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 3

([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3

([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 3

([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 3

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 3

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 3

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 3

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3

([(0,5),(1,4),(2,3)],6) => 3

([(1,5),(2,4),(3,4),(3,5)],6) => 3

([(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) => 4

([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 3

([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 3

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Description

The connected partition number of a graph.
This is the maximal number of blocks of a set partition $P$ of the set of vertices of a graph such that contracting each block of $P$ to a single vertex yields a clique.
Also called the pseudoachromatic number of a graph. This is the largest $n$ such that there exists a (not necessarily proper) $n$-coloring of the graph so that every two distinct colors are adjacent.

References

[1] van der Zypen, D. Connected partition number of a graph MathOverflow:381441
[2] Edwards, K. J. Achromatic number versus pseudoachromatic number: a counterexample to a conjecture of Hedetniemi DOI:10.1016/S0012-365X(00)00025-X

Code

def statistic(G, value_only=True):
    V = G.vertices()
    for k in range(len(V), 0, -1):
        for P in SetPartitions(V, k):
            H = G.copy(immutable=False)
            for B in P:
                H.merge_vertices(list(B))
            if H.is_clique():
                if value_only:
                    return k
                else:
                    return P

Created

Jan 17, 2021 at 18:00 by Martin Rubey

Updated

Mar 03, 2024 at 18:15 by Peter Kagey