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Number of unlabeled n-node graphs with the largest possible bipartite dimension (or biclique covering number).

2

`%I #11 Jul 02 2022 14:37:34
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`%S 1,1,1,6,7,5,1,372,326
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`%N Number of unlabeled n-node graphs with the largest possible bipartite dimension (or biclique covering number).
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`%C Let b(n) be the largest bipartite dimension of an n-node graph. Then b(n) >= A057359(n). If, for all n >= 8, there exists a disconnected n-node graph with bipartite dimension b(n), then b(n) = A057359(n) for all n >= 1. Proof: Since the bipartite dimension of a graph equals the sum of the bipartite dimensions of its connected components, we have that b(n) >= max_{k=1..n-1} b(k)+b(n-k) (i.e., the sequence is superadditive), with equality if there exists a disconnected n-node graph with bipartite dimension b(n). It is easily checked that A057359(n) = max_{k=1..n-1} A057359(k)+A057359(n-k) for n >= 8. Since b(n) = A057359(n) for 1 <= n <= 7 (checked by brute force), the result follows.
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`%C Since (b(n)) is superadditive, it follows from Fekete's subadditive lemma that the limit of b(n)/n exists and equals the supremum of b(n)/n. It is easy to see that b(n) <= b(n-1) + 1, so this limit is between 5/7 = b(7)/7 and 1.
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`%C The numbers of disconnected n-node graphs with bipartite dimension b(n) for 1 <= n <= 9 are 0, 0, 0, 2, 1, 1, 0, 12, 6.
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`%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bipartite_dimension">Bipartite dimension</a>
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`%Y Cf. A057359, A355334, A355335.
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`%K nonn,more
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`%O 1,4
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`%A _Pontus von BrÃ¶mssen_, Jun 29 2022
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