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Statistic identifier: St001345
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Collection: Graphs
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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 $k$.
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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]]
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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
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Statistic 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
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 5
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => 5
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 6
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 6
([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 4
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => 4
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => 5
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => 4
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => 5
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 5
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => 6
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => 5
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 6
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 6
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6) => 6
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 5
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 6
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 6
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 6
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 7
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 7
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 6
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 5
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 5
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 3
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6) => 3
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => 5
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 5
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 5
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 5
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 5
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 5
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 5
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => 6
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 6
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 6
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 6
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 5
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 6
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 6
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 6
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => 5
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6) => 5
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6) => 5
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 5
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6) => 4
([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 5
([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
([(0,3),(0,5),(1,2),(1,4),(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) => 4
([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 5
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6) => 6
([(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) => 4
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 4
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 4
([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 4
([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 5
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
([(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) => 5
([(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) => 5
([(0,3),(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
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 5
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 5
([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 5
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 3
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
([(0,2),(0,3),(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
([(0,1),(0,2),(0,3),(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) => 3
([],7) => 3
([(5,6)],7) => 5
([(4,6),(5,6)],7) => 4
([(3,6),(4,6),(5,6)],7) => 4
([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 5
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 6
([(3,6),(4,5)],7) => 5
([(3,6),(4,5),(5,6)],7) => 5
([(2,3),(4,6),(5,6)],7) => 4
([(4,5),(4,6),(5,6)],7) => 6
([(2,6),(3,6),(4,5),(5,6)],7) => 4
([(1,2),(3,6),(4,6),(5,6)],7) => 5
([(3,6),(4,5),(4,6),(5,6)],7) => 6
([(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 5
([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => 4
([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 6
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 6
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Created: Jan 16, 2019 at 22:36 by Martin Rubey
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Last Updated: Jan 21, 2019 at 11:28 by Martin Rubey