A213806 Number of minimal coprime labelings for the complete bipartite graph K_{n,n}.

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

Kevin Cuadrado, Table of n, a(n) for n = 1..105

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

Cf. A213273, A284875, A291465.

Sequence in context: A067616 A348992 A199377 * A019856 A124603 A199722

Adjacent sequences: A213803 A213804 A213805 * A213807 A213808 A213809

Keyword

nonn

Author

Alois P. Heinz, Jun 20 2012

Extensions

Terms a(24) and beyond from Kevin Cuadrado, Dec 01 2020

Status

approved