Computing the Grothendieck constant of some graph classes

The Grothendieck constant kappa(G) of a graph G = ([n], E) is the integrality gap of the canonical semidefinite relaxation of the integer program max(x is an element of[+/- 1]n) Sigma(ij is an element of E) w(ij)x(i) . x(j),variables by unit vectors. We show that kappa (G) = g/(g - 2) cos(pi/g) <= 3/2 when G has no K-5-minor and girth g; moreover, kappa(G) <= kappa(K-k) if the cut polyrope of G is defined by inequalities supported by at most k points; lastly the worst case ratio of clique-web inequalities is bounded by 3. (C) 2011 Elsevier B.V. All rights reserved.