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Institut FourierInstitut FourierUnit´e Mixte de Recherche 5582CNRS – Universit´e Joseph FourierOn the sharpness of some results relatingcuts and crossing numbers1 2Laurent Beaudou and Drago Bokal16 f´evrier 2009On the sharpness of some results relating cutsand crossing numbers1 2Laurent Beaudou and Drago Bokal1Institut Fourier100, rue des Maths38402 St-Martin d’H`eres – FRANCE2Faculty of Natural Science and MathematicsUniversity of MariborMaribor – SLOVENIAlaurent.beaudou@ujf-grenoble.frdrago.bokal@uni-mb.siAbstract/R´esum´eIt is already known that given two graphs together with two coherent bundlesin each graph, the crossing number of the zip-product of these graphs is greaterthan the sum of the crossing numbers of the graphs. When the size of thecut between the zipped graphs is at most two, just one bundle in each graphsuffices for the inequality. In this paper, we prove that such a relaxed conditionis not sufficient when the size of the cut is at least four, and we prove that thedifference can grow quadratically with the cut size.Keywords: crossing number, zip-product, graph cutEtant donn´es deux graphes contenant deux bundles coh´erents, on sait par unth´eor`eme du deuxi`eme auteur que le nombre de croisements du produit zipp´edes deux graphes est plus grand que le la somme des nombres de croisementsde chaque graphe. Dans ce travail, nous r´epondons n´egativement `a la questionnaturelle de la n´ecessit´e de ces deux bundles dans chaque ...