OFFSET
2,3
COMMENTS
The difference between "fundamentally different graceful labelings" of a graph and "graceful labelings" of a graph is that the latter is the former multiplied by twice the number of automorphisms. (The extra factor of 2 comes from complementation.)
a(10) >= 127055. - Eric W. Weisstein, Dec 19 2025
REFERENCES
D. E. Knuth, The Art of Computer Programming, Section 7.2.2.3, in preparation.
LINKS
Eric Weisstein's World of Mathematics, Graceful Labeling.
Eric Weisstein's World of Mathematics, Maximally Graceful Graph.
EXAMPLE
For n=4 the "paw" graph has a(4)=5 fundamentally different labelings, namely with edges
0-4,0-3,0-2,2-3 or
0-4,0-3,0-2,3-4 or
0-4,0-3,1-3,0-1 or
0-4,0-3,1-3,3-4 or
0-4,0-3,2-4,3-4.
The other six graphs with four vertices are either ungraceful (2K_1) or uniquely graceful (K_1,3, K_4, C_4, P_4) or have fewer than 5 (K_1,1,2 has 4).
For n=5 the "dart" has a(5)=26 fundamentally different labelings.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Don Knuth, Dec 21 2020
EXTENSIONS
a(9) from Eric W. Weisstein, Nov 11 2025
a(10) from Eric W. Weisstein, Feb 27 2026
STATUS
approved
