A363364 -id:A363364 - cp's OEIS frontend

A363362 Number of connected weakly pancyclic graphs on n unlabeled nodes.

1, 1, 2, 6, 21, 108, 827, 10908, 259567, 11704426

Offset: 1

Pontus von Brömssen, May 29 2023

A graph is weakly pancyclic if it contains cycles of all lengths between its girth and its circumference. Acyclic graphs are considered to be weakly pancyclic. The concept of weak pancyclicity was introduced by Brandt, Faudree, and Goddard (1998).

Stephan Brandt, Ralph Faudree, and Wayne Goddard, Weakly pancyclic graphs, Journal of Graph Theory 27 (1998), 141-176.

Cf. A001349, A286684, A363363, A363364.

a(n) = A001349(n) - A363363(n).

a(n) = A001349(n) for n <= 5, because all graphs on at most 5 nodes are weakly pancyclic.

A363363 Number of connected unlabeled n-node graphs G that are not weakly pancyclic, i.e., there exists an integer k such that G contains a cycle that is longer than k and a cycle that is shorter than k but no cycle of length k.

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 26, 209, 1513, 12145

Offset: 1

Pontus von Brömssen, May 29 2023

			There are a(6) = 4 not weakly pancyclic graphs on 6 nodes (all of them connected):
  a cycle of length 6 with one additional edge (two different graphs);
  the complete bipartite graph K_{3,3} with one edge removed;
  K_{3,3}.

Stephan Brandt, Ralph Faudree, and Wayne Goddard, Weakly pancyclic graphs, Journal of Graph Theory 27 (1998), 141-176.

Cf. A001349, A363362, A363364.

a(n) = A001349(n) - A363362(n).

a(n) = 0 for n <= 5, because all graphs on at most 5 nodes are weakly pancyclic.

cp's OEIS Frontend

A363362 Number of connected weakly pancyclic graphs on n unlabeled nodes.

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