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A213806
Number of minimal coprime labelings for the complete bipartite graph K_{n,n}.
4
1, 1, 7, 3, 1, 3, 4, 5, 1, 9, 1, 1, 39, 2, 46, 16, 42, 68, 1, 175, 1, 5, 50, 1, 627, 1256, 1177, 10, 1860, 7144, 15, 170, 27156, 178, 64, 2, 6335, 6334, 15592, 4522, 3230, 113926, 99010, 72256, 114606, 199042, 1, 198518, 151036, 236203, 8557, 26542, 21388
OFFSET
1,3
COMMENTS
A minimal coprime labeling for K_{n,n} uses two disjoint n-subsets of {1,...,m} with minimal m = A213273(n) >= 2*n as labels for the two disjoint vertex sets such that labels of adjacent vertices are relatively prime. One of the label sets contains m.
LINKS
Adam H. Berliner, N. Dean, J. Hook, A. Marr, A. Mbirika, C. McBee, Coprime and prime labelings of graphs, arXiv preprint arXiv:1604.07698 [math.CO], 2016.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
FORMULA
a(A284875(n)) = 1. - Jonathan Sondow, May 21 2017
EXAMPLE
a(1) = 1: the two label sets are {{1}, {2}} with m=2.
a(2) = 1: {{1,3}, {2,4}} with m=4.
a(3) = 7: {{2,4,5}, {1,3,7}}, {{1,3,5}, {2,4,7}}, {{2,3,4}, {1,5,7}}, {{2,3,6}, {1,5,7}}, {{2,4,6}, {1,5,7}}, {{3,4,6}, {1,5,7}}, {{1,2,4}, {3,5,7}}.
a(4) = 3: {{2,4,7,8}, {1,3,5,9}}, {{2,4,5,8}, {1,3,7,9}}, {{1,2,4,8}, {3,5,7,9}}.
a(5) = 1: {{2,4,5,8,10}, {1,3,7,9,11}}.
a(21) = 1: {{2,4,5,8,10,11,16,20,22,23,25,29,31,32,40,44,46,50,55,58,62}, {1,3,7,9,13,17,19,21,27,37,39,41,43,47,49,51,53,57,59,61,63}}.
MAPLE
b:= proc(n, k, t, s) option remember;
`if`(nops(s)>=t and k>=t, binomial(nops(s), t),
`if`(n<1, 0, b(n-1, k, t, s)+ b(n-1, k+1, t,
select(x-> x<>n and igcd(n, x)=1, s))))
end:
g:= proc(n) option remember; local m, r;
for m from `if`(n=1, 2, g(n-1)[1]) do
r:= b(m-1, 1, n, select(x-> igcd(m, x)=1, {$1..m-1}));
if r>0 then break fi
od; [m, r]
end:
a:= n-> g(n)[2]:
seq(a(n), n=1..11);
MATHEMATICA
b[n_, k_, t_, s_] := b[n, k, t, s] = If[Length[s] >= t && k >= t, Binomial[Length[s], t], If[n < 1, 0, b[n - 1, k, t, s] + b[n - 1, k + 1, t, Select[s, # != n && GCD[n, #] == 1 &]]]];
g[n_] := g[n] = Module[{m, r}, For[ m = If[n == 1, 2, g[n - 1][[1]] ], True, m++, r = b[m - 1, 1, n, Select[Range[1, m - 1], GCD[m, #] == 1 &]]; If [r > 0, Break[]]]; {m, r}];
a[n_] := a[n] = g[n][[2]];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 18}] (* Jean-François Alcover, Nov 08 2017, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 20 2012
EXTENSIONS
Terms a(24) and beyond from Kevin Cuadrado, Dec 01 2020
STATUS
approved