The Grothendieck constant κ(G) of a graph G=([n],E) is the integrality gap of the canonical semidefinite relaxation of the integer program maxx∈±1n∑ij∈Ewijxi·xj, replacing ±1 variables by unit vectors. We show that κ(G)=g(g-2)cos(πg)≤32 when G has noK5-minor and girth g; moreover, κ(G)≤κ(Kk) if the cut polytope 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. © 2011 Elsevier B.V. All rights reserved.
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