OFFSET
1,3
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
a(10) >= 105.
LINKS
Fedor V. Fomin, Serge Gaspers, Artem V. Pyatkin, and Igor Razgon, On the minimum feedback vertex set problem: exact and enumeration algorithms, Algorithmica 52 (2008), 293-307.
Natasha Morrison and Alex Scott, Maximising the number of induced cycles in a graph, Journal of Combinatorial Theory Series B 126 (2017), 24-61.
FORMULA
a(m+n) >= a(m)*a(n).
1.5926... = 105^(1/10) <= lim_{n->oo} a(n)^(1/n) <= 1.8638. (Fomin, Gaspers, Pyatkin, and Razgon (2008)).
EXAMPLE
All optimal graphs (i.e., graphs having n nodes and a(n) maximal acyclic 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 Fomin, Gaspers, Pyatkin, and Razgon, 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, 2, 2, 2);
n = 8: FCB(1, 2, 1, 2, 2);
n = 9: FCB(1, 2, 2, 1, 3).
CROSSREFS
Sequences of largest number of maximal induced subgraphs with a given property:
A000792 (independent sets or cliques),
this sequence (acyclic),
A342212 (bipartite),
A342213 (planar),
A342324 (chordal),
A352208 (3-colorable),
A352209 (perfect),
A352210 (2-degenerate),
A352211 (cluster graphs),
A352212 (triangle-free),
A352213 (cographs),
A352214 (claw-free),
A352215 (C_4-free),
A352216 (diamond-free).
KEYWORD
nonn,more
AUTHOR
Pontus von Brömssen, Mar 05 2021.
STATUS
approved