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Identifier

St001707: Graphs ⟶ ℤ

Values

([],1) => 1

([],2) => 0

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

([],3) => 1

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

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

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

([],4) => 0

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

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

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

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

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

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

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

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

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

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

([],5) => 1

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

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

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

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

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

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

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

([(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) => 5

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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) => 5

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

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

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

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

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

([],6) => 0

([(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) => 4

([(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) => 4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them.
Such a partition always exists because of a construction due to Dudek and Pralat [1] and independently Pokrovskiy [2].

References

[1] Dudek, A., Prałat, Paweł An alternative proof of the linearity of the size-Ramsey number of paths MathSciNet:3326432
[2] Pokrovskiy, A. Partitioning edge-coloured complete graphs into monochromatic cycles and paths MathSciNet:3194196

Code

def statistic(G):
    n = G.num_verts()
    for k in reversed(range(n+1)):
        if is_odd(n-k):
            continue
        l = (n-k)//2
        for P in Subsets(G.vertices(), k):
            V = set(G.vertices())
            for U in Subsets(V.difference(P), l):
                if all(all(v in U or v in P for v in G[u]) for u in U):
                    if len(P) <= 1 or G.subgraph(P).hamiltonian_path():
                        return k

Created

Apr 03, 2021 at 16:12 by Martin Rubey

Updated

Apr 03, 2021 at 16:12 by Martin Rubey