***************************************************************************** * 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: St001345 ----------------------------------------------------------------------------- Collection: Graphs ----------------------------------------------------------------------------- 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 ----------------------------------------------------------------------------- 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 ----------------------------------------------------------------------------- Created: Jan 16, 2019 at 22:36 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Jan 21, 2019 at 11:28 by Martin Rubey