A245231 Maximum frustration of complete bipartite graph K(n,4).

0, 2, 3, 4, 5, 7, 8, 10, 10, 12, 13, 14, 15, 17, 18, 20, 20, 22, 23, 24, 25, 27, 28, 30, 30, 32, 33, 34, 35, 37, 38, 40, 40, 42, 43, 44, 45, 47, 48, 50, 50, 52, 53, 54, 55, 57, 58, 60, 60, 62, 63, 64, 65, 67, 68, 70, 70, 72, 73, 74, 75, 77, 78, 80, 80, 82, 83, 84, 85, 87, 88, 90, 90, 92, 93, 94, 95, 97, 98, 100

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, A245227, A245239, A245314.

Programs

Maple

A:= n -> floor(5*n/4) - piecewise(member(n mod 8, {1,4,5}),1,0);
seq(A(n),n=1..100);

Mathematica

a[n_] := Floor[5n/4] - If[MemberQ[{1, 4, 5}, Mod[n, 8]], 1, 0];
Array[a, 100] (* Jean-François Alcover, Mar 28 2019, from Maple *)

Formula

a(n) = floor(5*n/4) - 1 if n == 1, 4 or 5 mod 8, a(n) = floor(5*n/4) otherwise.

G.f.: x^2*(2*x^6+x^5+2*x^4+x^3+x^2+x+2)/(x^9-x^8-x+1).

a(n+8) = a(n) + 10.

a(n) = A245230(max(n,4), min(n,4)).