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.)
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
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
STATUS
approved