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Identifier

St001345: Graphs ⟶ ℤ

Values

([],1) => 0

([],2) => 1

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

([],3) => 2

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

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

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

([],4) => 2

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

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

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

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

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

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

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

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

([(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) => 2

([],5) => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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) => 6

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

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

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

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

([(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) => 4

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

([(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) => 5

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

([(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) => 3

([],6) => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 7

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

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

([(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) => 5

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

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

([(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) => 5

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

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

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 7

([(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) => 6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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) => 5

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

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Description

The Hamming dimension of a graph.
Let $H(n, k)$ be the graph whose vertices are the subsets of $\{1,\dots,n\}$, and $(u,v)$ being an edge, for $u\neq v$, if the symmetric difference of $u$ and $v$ has cardinality at most $k$.
This statistic is the smallest $n$ such that the graph is an induced subgraph of $H(n, k)$ for some $k$.

References

[1] van der Zypen, D. Graph embeddings into Hamming spaces van der Zypen, D. Graph embeddings into Hamming spaces arXiv:1901.03409
[2] van der Zypen, D. Hamming representability of finite graphs van der Zypen, D. Hamming representability of finite graphs MathOverflow:319951

Code

def Hamming(n, k):
    V = [frozenset(v) for v in subsets(range(n))]
    return Graph([V, lambda a,b: 0 < len(a.symmetric_difference(b)) <= k])

@cached_function
def statistic(G):
    n = -1
    while True:
        n += 1
        V = [frozenset(v) for v in subsets(range(n))]
        for k in range(n+1):
            H = Graph([V, lambda a,b: 0 < len(a.symmetric_difference(b)) <= k])
            if H.subgraph_search(G, induced=True):
                return n

Created

Jan 16, 2019 at 22:36 by Martin Rubey

Updated

Jan 21, 2019 at 11:28 by Martin Rubey