A379575 Total numbers of fundamentally distinct graceful labelings among all simple graphs on n vertices having no isolated points.

0, 1, 2, 13, 157, 3292, 110578, 5903888, 485522560, 60576177550

Offset

1,3

References

Knuth, D.E. Problem 97 in The Art of Computer Programing, Vol. 4, Section 7.2.2.3. In preparation.

Links

Table of n, a(n) for n=1..10.

Eric Weisstein's World of Mathematics, Graceful Labeling.

Eric Weisstein's World of Mathematics, Isolated Point.

Example

In the below, G: n stands for "G has n fundamentally distinct graceful labelings".
a(1) = 0 since K_1 is an isolated point.
a(2) = 1 since P_2: 1.
a(3) = 2 since P_3: 1, C_3: 1.
a(4) = 13 since K_1,3 (claw): 1, diamond: 4, P_4: 1, paw: 5, C_4: 1, K_4: 1, and C_3+K_1: 0 (since it contains an isolated point).

Mathematica

{0, 1} ~ Join ~ Table[Total[GraphData[#, "GracefulLabelingCount"] & /@ Select[GraphData["Graceful", n], GraphData[#, "MinimumVertexDegree"] > 0 &]], {n, 3, 7}]

Crossrefs

Cf. A333727 (totals of all graceful labelings of simple graphs on n vertices).

Cf. A379576 (totals of all fundamentally distinct graceful labelings of simple graphs on n vertices).

Sequence in context: A377571 A380719 A316701 * A062593 A347051 A291140

Adjacent sequences: A379572 A379573 A379574 * A379576 A379577 A379578

Keyword

nonn,more,hard

Author

Eric W. Weisstein, Dec 26 2024

Extensions

a(9) from Eric W. Weisstein, Nov 12 2025

a(10) from Eric W. Weisstein, Feb 27 2026

Status

approved