A245227 Maximum frustration of complete bipartite graph K(n,5).

0, 2, 3, 5, 7, 9, 10, 12, 13, 15, 17, 18, 19, 21, 22, 25, 26, 27, 29, 30, 32, 34, 35, 37, 38, 40, 42, 43, 44, 46, 47, 50, 51, 52, 54, 55, 57, 59, 60, 62, 63, 65, 67, 68, 69, 71, 72, 75, 76, 77, 79, 80, 82, 84, 85, 87, 88, 90, 92, 93, 94, 96, 97, 100, 101, 102

Offset: 1

Author

Robert Israel, Jul 14 2014

Keywords

• nonn

Comments

The maximum frustration of a graph is the maximum cardinality of a set of edges that contains at most half the edges of any cut-set. Another term that is used is "line index of imbalance". It is also equal to the covering radius of the coset code of the graph.

Examples

			For n=2 a set of edges that attains the maximum cardinality a(2)=2 is {(1,3),(1,4)}.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• G. S. Bowlin, Maximum Frustration in Bipartite Signed Graphs, Electr. J. Comb. 19(4) (2012) #P10.
• R. L. Graham and N. J. A. Sloane, On the Covering Radius of Codes, IEEE Trans. Inform. Theory, IT-31(1985), 263-290.
• P. Solé and T. Zaslavsky, A Coding Approach to Signed Graphs, SIAM J. Discr. Math 7 (1994), 544-553.

Crossrefs

Cf. A245230, A245231, A245239, A245314.

Programs

Maple

A245227:= n -> floor(25/16*n) - piecewise(member(n mod 16, {2,4,9,13,15}),1,0):
A245227(1):= 0:
A245227(3):= 3:
seq(A245227(n),n=1..100);

Mathematica

a[n_] := Floor[25 n/16] - If[n == 1 || n == 3 || MemberQ[{2, 4, 9, 13, 15}, Mod[n, 16]], 1, 0];
Array[a, 100] (* Jean-François Alcover, Mar 27 2019, after Robert Israel *)

Formula

a(n) = floor(25/16*n) - 1 if n == 2,4,9,13, or 15 mod 16 or if n = 1 or 3; a(n) = floor(25/16*n) otherwise.

G.f.: -x^2*(x^18-x^17+x^16-x^15-3*x^14-x^13-2*x^12-x^11-x^10-2*x^9-2*x^8-x^7-2*x^6-x^5-2*x^4-2*x^3-2*x^2-x-2)/(x^17-x^16-x+1).

a(n+16) = a(n) + 25 for n > 3.

a(n) = A245230(max(n,5),min(n,5)).