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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 ...

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Fourier
Institut Fourier
Unit´eMixtedeRecherche5582 CNRSUniversit´eJosephFourier
On the sharpness of some results relating cuts and crossing numbers
1 2 Laurent Beaudou and Drago Bokal
16f´evrier2009
On the sharpness of some results relating cuts and crossing numbers
1 2 Laurent Beaudou and Drago Bokal
1 Institut Fourier 100, rue des Maths 38402StMartindH`eresFRANCE 2 Faculty of Natural Science and Mathematics University of Maribor Maribor – SLOVENIA
laurent.beaudou@ujfgrenoble.fr drago.bokal@unimb.si
Abstract/Re´sum´e
It is already known that given two graphs together with two coherent bundles in each graph, the crossing number of the zipproduct of these graphs is greater than the sum of the crossing numbers of the graphs. When the size of the cut between the zipped graphs is at most two, just one bundle in each graph suffices for the inequality. In this paper, we prove that such a relaxed condition is not sufficient when the size of the cut is at least four, and we prove that the difference can grow quadratically with the cut size. Keywords:crossing number, zipproduct, graph cut
Etantdonn´esdeuxgraphescontenantdeuxbundlescohe´rents,onsaitparun the´ore`medudeuxie`meauteurquelenombredecroisementsduproduitzipp´e des deux graphes est plus grand que le la somme des nombres de croisements dechaquegraphe.Danscetravail,nousre´pondonsn´egativement`alaquestion naturelledelane´cessite´decesdeuxbundlesdanschaquegraphe.Deplus,nous exhibonsunefamilledegraphere´alisantunedie´rencearbitrairementgrande entrecesdeuxquantite´s Motscle´s:crossing number.
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Introduction
Crossing number of graphs (see [11] for basic definitions) has been extensively studied for about thirty years and is still a notorious problem in graph theory. While determining or bounding the crossing number of graphs used to be the main issue at the beginning, the focus is now shifting to structural aspects of the crossing number problem. These include the study of several variants of crossing number [5, 7, 15, 16, 18, 19], crossingcritical graphs [8, 20, 21], or the properties of drawings with a bounded number of crossings per edge [17]. Very early at the development of the crossing number theory Leighton re alized that cuts in graphs play an important role in determining the crossing number of a graph. Combining them with the LiptonTarjan planar separator theorem [14], he used edge cuts in graphs to provide upper bounds on cross ing numbers [1], whereas in the bisection method [12, 13], he used this fact to provide lower bounds for the crossing number. Both the upper and the lower bound arising from graph cuts are rather gen eral methods, but neither provides the sharpness needed to yield exact bounds. This issue was resolved by the introduction of the zip product of graphs in [2, 3], which led to exact crossing number of several twoparameter graph fam ilies, most general being the crossing number of the Cartesian product of any subcubic tree with any starK1,nfamily includes, as a subfamily, the. This product of any path and any star, resolving a longstanding conjecture by Jen ´ ˇ drolandScˇerbovain[10].Besides,ithasalsobeenhelpfulforotherworks concerning exact crossing numbers (as in [22, 23]), but also regarding crossing critical graphs (see [4, 9]). It is natural to ask about its behaviour with respect to graph invariants. The zip product approach, however, assumes a technical condition of having two coherent bundles in the zipped graphs (we formalize this condition later). In this paper, we examine the possible weakenings of this condition and prove that having only one bundle in each graph is not sufficient to establish superaditivity of crossing number with regard to the zip product. Moreover, we are able to achieve any gap between both quantities and show that the gap can grow quadratically with the number of edges involved in the zip product. In the first section, we state definitions of zip product and bundles and recall precedent results. In the second part, we describe the families of graphs that we use in the proof of our main result.
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The zip product
Fori= 1,2, letGibe a graph andviV(Gi) such that bothv1andv2have same degreed. LetNi=NGi(vi) be the set of neighbouring vertices ofviin Gi, and letσ:N1N2We callbe a bijection. σazip functionof the graphs G1andG2at verticesv1andv2. Thezip productofG1andG2according toσ is the graphG1σG2obtained from the disjoint union ofG1v1andG2v2 after adding edges(u) for anyuN1. LetvV(G) be a vertex of degreedinG. Abundleofvis a setBofd edgedisjoint paths fromvto some vertexuV(G),u6=v. Vertexvis the sourceof the bundle anduis itssinkvertices on the paths of. Other Bare ˘ internal verticesof the bundle. LetE(B) =E(B)E(Gv) denote the set of
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edges of bundles The
Bthat are not incident withv. They are B1andB2ofvarecoherentif their sets following result was established in [3]:
calleddistant edgesofB. Two of distant edges are disjoint.
Theorem 2.1(Bokal [3])Fori= 1,2, letGibe a graph,viV(Gi)a vertex of degreed, andNi=NGi(vi). Also assume thatvihas two coherent bundles Bi,1andBi,2inGi. Then,cr(G1σG2)cr(G1) + cr(G2)for any bijection σ:N1N2.
Note that, as stated above, the zip product is defined to involve vertices that have no incident multiple edges. However, the reader shall have no difficulty establishing the same result for graphs with multiple edges: if the edges in each multiple edge are subdivided, the crossing number is preserved and the resulting graph is simple. The zip product is done using these simple graphs. If necessary, we can suppress the degreetwo edges after the zip product, and obtain back the multiple edges.
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Two families of graphs
We define two families of graphs with specific crossing number, such that in each a chosen vertex has a single bundle.
Definition 3.1Given any integern4, letn= 4k+r. LetK=K2,4with bipartitionai,i= 1,2andbj,j= 1,2,3,4. The graphHk,ris obtained fromK with the following steps:
1. adding a cycleC=b1b2b3b4, 2. subdividing the edgea1biwithxi,i= 1,3, 3. adding the edgex1x3, 4. replacing every edge withkparallel edges,
5. addingrmore edges to the multiedgesaib4,i= 1,2.
Figure 1(a) depictsH3,1.
2 Lemma 3.2Fork1andr∈ {0,1,2,3},cr(Hk,r) =k.
Proof.It is easy to find a subdivision ofK3,3inH1,r, thus cr(H1,r)1. Figure 1(a) gives a natural way of drawingH1,0with 1 crossing, establishing 2 2 cr(H1,0Furthermore, it is obvious that cr() = 1. Hk,0) =kcr(H1,0) =k. Since Hk,0is a subgraph ofHk,r, we have cr(Hk,r)cr(Hk,0the other hand,). On one can alter an optimal drawing ofHk,0to a drawing ofHk,rwith the same 2 number of crossings (as in Figure 1(a), concluding cr(Hk,r) = cr(Hk,0) =k.
Definition 3.3Given any integern4, letn= 4k+r. LetK=K2,4with bipartitionai,i= 1,2andbj,j= 1,2,3,4graph. The Gk,ris obtained fromK with the following steps:
1. adding a cycleC=b1b2b3b4,
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b4
b3
x3
a1
a2
b2
(a) GraphH3,1
x1
b1
b4
x3 y3 b3
a1
a2
x1 y1 b2
(b) GraphG3,1
Figure 1: Two families of graphs:Gk,randHk,r
2. subdividing the edgea1bitwice, withxi,yi,i= 1,3,
3. adding the edgesx1y3andx3y1,
4. replacing every edge withkparallel edges,
5. addingrmore edges to the multiedgesaib4,i= 1,2.
Figure 1(b) depictsG3,1.
2 Lemma 3.4Fork1andr∈ {0,1,2,3},cr(Gk,r) = 2k.
b1
Proof.The subgraphG1,0contains a subdivision of a graph, obtained from K3,4by splitting two vertices of degree 4. This graph has crossing number two [6], and the drawing in Figure 1(b) gives a way of drawingG1,0establishing 2 2 cr(G1,0) = 2. By construction, cr(Gk,0) =kcr(G1,0) = 2k. Furthermore, it is easy to modify the optimal drawing ofGk,0to a drawing ofGk,rwith the same number of crossings as in Figure 1(b). AsGk,rhasGk,0as a subgraph, we have 2 cr(Gk,r) = cr(Gk,0) = 2k.
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The Zip Product Gap
Proposition 4.1LetG1,G2be graphs,viV(Gi)of degreedhas one bundle BiinGi,σa bijection among neighborhoods ofvi. If there exists an optimal drawingDofG1σG2, such that no edge ofG1v1crosses an edge ofG2v2, thencr(G1σG2)cr(G1) + cr(G2).
Proof.Without loss of generality we may assume thatGi,i= 1,2 are con nected. Due to the bundles,Giviare connected, too. LetDibe obtained from D[(Givi)B3i] by contracting any of its faces whose boundary contains only
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segments of edges ofB3i. ThenDiis a drawing ofGi, with the contracted region representing the vertexvi. Clearly, each crossing ofDiappears inD, thus cr(G1σG2) = cr(D)cr(D1) + cr(D2)cr(G1) + cr(G2).
The following theorem establishes that in general, one bundle at each vertex used in the zip product does not suffice for preservation of the crossing number: Theorem 4.2For anyn4, there exist graphsG1,G2, such thatGihas a vertexviwith a bundle inGi,dGi(vi) =n, andcr(G1v1v2G2)<cr(G1) + cr(G2). Proof.Letn= 4k+rand setG1=Hk,r,G2=Gk,r, and letvi,i= 1,2, 2 be the vertexa1from the definition of the respective graph. Then cr(G1) =k, 2 cr(G2) = 2k.
(a)G1,1H1,1
(b) Better drawing ofG1,1H1,1
Figure 2:G1,1H1,1
2 2 The drawing in Figure 2(b) establishes cr(G1v1v2G2)2k <3k= cr(G1) + cr(G2).
Combining Theorem 4.2 with Proposition 4.1, we obtain the following: Corollary 4.3For anyn4, there exist graphsG1,G2, such thatGihas a vertexviwith a bundle,dGi(vi) =n, andG1v1crossesG2v2in any optimal drawing ofG1v1v2G2. Let graphsG1andG2bencompatible, if eachGicontains a vertexviof degreen, such thatvihas a bundle inGi. Define g(nmax [cr() = G) + cr(G)cr(GG)], 1 2 1v1v22 G1,G2
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where the maximum runs over allncompatible pairsG1,G2. Theorem 4.2 establishes the following corollary:
2 Corollary 4.4Letg(n)Thenbe defined as above. g(n) = Ω(n).
The proof of
Thus, we have a lower bound on the possible crossing number gap between the two original graphs and their zip product in terms of the size of the edge cut between the zipped graphs. An upper bound, on the other hand, is far from clear, as the edges of the two bundles can possibly cross each other arbitrarily often in the optimal drawing of the zip product, and optimal drawings of the original graphs can have different structure than the corresponding subdrawings of optimal drawings of the zipped graph. We summarize this discussion in the following problem:
2 Problem 4.5Find an upper bound ong(n). Isg(n) =O(n)?
Theorem 2.1 establishes that if the vertices involved in the zip product have two coherent bundles, then the crossing number is preserved. On the other hand, Theorem 4.2 establishes that whenever their degree is at least four, just one bundle at each vertex is not enough. Therefore, two natural problems remain open:
Problem 4.6LetG1,G2be graphs, such thatGihas a vertexviwith a bundle inGianddGi(vi) = 3. Iscr(G1v1v2G2)cr(G1) + cr(G2)?
Problem 4.7LetG1,G2be graphs, such thatGihas a vertexviwithdGi(vi) = dthat. Assume v1has two coherent bundles inG1, butv2has just one bundle inG2. Iscr(G1v1v2G2)cr(G1) + cr(G2)?
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