A335619 - OEIS

%I #22 Dec 08 2020 21:59:59

%S 1,1,4,1,7,2,10,3,8,1,42,2,7,7

%N Number of fundamentally different graceful labelings of the complete bipartite graph K_{n,n}.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GracefulLabeling.html">Graceful Labeling</a>

%F a(n) = A337793(n)/(4*(n!)^2).

%e a(3) = 4 because there are 4 fundamentally different graceful labelings:

%e solution #1:

%e    0  1  4  5

%e    6  8 14 16

%e solution #2:

%e    0  1  8  9

%e   10 12 14 16

%e solution #3:

%e    0  1  2 15

%e    5  9 13 16

%e solution #4

%e    0  1  2  3:

%e    4  8 12 16

%e All others can be obtained by permutations of left and right vertices, swapping halves ("0" vertex left or right) and the replacement of all vertex labels k by N^2-k. - noted by _Bert Dobbelaere_, Oct 01 2020

%Y Cf. A337793 (total number of graceful labelings).

%K nonn,more

%O 2,3

%A _Eric W. Weisstein_, Oct 02 2020

%E a(10)-a(15) from the diagonal of A337278 by _Don Knuth_, Dec 08 2020