

A355336


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


2




OFFSET

1,4


COMMENTS

Let b(n) be the largest bipartite dimension of an nnode graph. Then b(n) >= A057359(n). If, for all n >= 8, there exists a disconnected nnode 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..n1} b(k)+b(nk) (i.e., the sequence is superadditive), with equality if there exists a disconnected nnode graph with bipartite dimension b(n). It is easily checked that A057359(n) = max_{k=1..n1} A057359(k)+A057359(nk) for n >= 8. Since b(n) = A057359(n) for 1 <= n <= 7 (checked by brute force), the result follows.
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(n1) + 1, so this limit is between 5/7 = b(7)/7 and 1.
The numbers of disconnected nnode graphs with bipartite dimension b(n) for 1 <= n <= 9 are 0, 0, 0, 2, 1, 1, 0, 12, 6.


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KEYWORD

nonn,more


AUTHOR



STATUS

approved



