A308722 Number of edges in the smallest possible regular graceful graph of valence n.

0, 1, 3, 6, 12, 20, 27, 35, 48, 63, 75, 99, 108

Offset

0,3

Comments

A graceful graph with e edges is one that allows a labeling of its vertices with distinct integers between 0 and e, inclusive, such that when each edge is labeled with the absolute difference of the labels of its endpoints, these edge labels will be exactly the integers between 1 and e, inclusive.

This sequence counts the minimal number edges possible, in a regular graceful graph of valence n.

The concept was introduced by Ed Pegg Jr in a post on Mathematics Stack Exchange; all the data here is from that post.

Pegg conjectures that the sequence continues (from n = 13): 130, 147, 180, 192, 221, 243, 285.

Links

Table of n, a(n) for n=0..12.

Mathematics Stack Exchange, Graceful graphs with Valence k

Formula

Conjecture: a(2*n) = 3*n^2 = A033428(n). - Ed Pegg Jr, Dec 11 2024

Example

See the link at Mathematics Stack Exchange.

Crossrefs

Sequence in context: A181962 A226220 A218076 * A320192 A160732 A320608

Adjacent sequences: A308719 A308720 A308721 * A308723 A308724 A308725

Keyword

nonn,more

Author

Allan C. Wechsler, Jun 19 2019

Extensions

More terms from Ed Pegg Jr, Jul 17 2019

Status

approved