A342212 Largest number of maximal bipartite node-induced subgraphs of an n-node graph.

1, 1, 3, 6, 10, 15, 21, 38, 64

Offset: 1

Author

Pontus von Brömssen, Mar 05 2021

Keywords

• nonn
• more

Comments

This sequence is log-superadditive, i.e., a(m+n) >= a(m)*a(n). By Fekete's subadditive lemma, it follows that the limit of a(n)^(1/n) exists and equals the supremum of a(n)^(1/n). - Pontus von Brömssen, Mar 03 2022

Byskov, Madsen, and Skjernaa (2005) construct a 10-node graph with 105 maximal bipartite subgraphs, so a(10) >= 105.

Examples

			All optimal graphs (i.e., graphs having n nodes and a(n) maximal bipartite subgraphs) for 1 <= n <= 9 are listed below. Here, FCB(n_1, ..., n_k) denotes the full cyclic braid graph with cluster sizes n_1, ..., n_k, as defined by Morrison and Scott (2017), i.e., the graph obtained by arranging complete graphs of orders n_1, ..., n_k (in that order) in a cycle, and joining all pairs of nodes in neighboring parts with edges. (The graph in the paper by Byskov, Madsen, and Skjernaa, which shows that a(10) >= 105, is FCB(2, 2, 2, 2, 2).)
        n = 1: the 1-node graph;
        n = 2: the complete graph and the empty graph;
  3 <= n <= 6: the complete graph;
        n = 7: FCB(1, 1, 2, 1, 2) (the Moser spindle) and the complete graph;
        n = 8: FCB(1, 2, 1, 2, 2) and the 4-antiprism graph;
        n = 9: FCB(1, 2, 2, 1, 3).
		

Links

• Jesper Makholm Byskov, Bolette Ammitzbøll Madsen, and Bjarke Skjernaa, On the number of maximal bipartite subgraphs of a graph, Journal of Graph Theory 48 (2005), 127-132.
• Natasha Morrison and Alex Scott, Maximising the number of induced cycles in a graph, Journal of Combinatorial Theory Series B 126 (2017), 24-61.

Crossrefs

Cf. A005142, A033995.

For a list of related sequences, see cross-references in A342211.

Formula

a(m+n) >= a(m)*a(n).

a(n) <= n*12^(n/4). (Byskov, Madsen, and Skjernaa (2005))

1.5926... = 105^(1/10) <= lim_{n->oo} a(n)^(1/n) <= 12^(1/4) = 1.8612... . (Byskov, Madsen, and Skjernaa (2005))