Making a graph crossing-critical by multiplying its edges
aa r X i v : . [ m a t h . C O ] D ec Making a graph crossing-criticalby multiplying its edges
Laurent Beaudou , C´esar Hern´andez–V´elez , and Gelasio Salazar ∗ ,2 LIMOS, Universit´e Blaise Pascal, Clermont-Ferrand, France Instituto de F´ısica, Universidad Aut´onoma de San Luis Potos´ı. San Luis Potos´ı, Mexico 78000
November 10, 2018
Abstract
A graph is crossing-critical if the removal of any of its edges decreases its crossingnumber. This work is motivated by the following question: to what extent is crossing-criticality a property that is inherent to the structure of a graph, and to what extentcan it be induced on a noncritical graph by multiplying (all or some of) its edges?It is shown that if a nonplanar graph G is obtained by adding an edge to a cubicpolyhedral graph, and G is sufficiently connected, then G can be made crossing-criticalby a suitable multiplication of edges. This work is motivated by the recent breakthrough constructions by DeVos, Mohar andˇS´amal [3] and Dvoˇr´ak and Mohar [5], which settled two important crossing numbers ques-tions. The graphs constructed in [3] and [5] use weighted (or “thick”) edges. A graphwith weighted edges can be naturally transformed into an ordinary graph by substitutingweighted edges by multiedges (recall that a multiedge is a set of edges with the same pair ofendvertices). If one wishes to avoid multigraphs, one can always substitute a weight t edgeby a K ,t , but still the resulting graph is homeomorphic to a multigraph. Sometimes (asin [3]) one can afford to substitute each weighted edge by a slightly richer structure (such asa graph obtained from K ,t by joining the degree 2 vertices with a path), but sometimes (asin [5]) one is concerned with criticality properties, and so no such superfluous edges may beadded. In any case, the use of weighted edges is crucial.After trying unsuccessfully to come up with graphs with similar crossing number proper-ties as those presented in [3] and [5], while avoiding the use of weighted edges, we were left ∗ Supported by CONACYT Grant 106432. crossing number cr( G ) of a graph G is the minimum number of pairwiseintersections of edges in a drawing of G in the plane. An edge e of G is crossing-critical ifcr( G − e ) < cr( G ). If all edges of G are crossing-critical, then G itself is crossing-critical . Acrossing-critical graph seems naturally more interesting than a graph with some not crossing-critical edges, since a graph of the latter kind contains a proper subgraph that has all therelevant information from the crossing numbers point of view.Earlier constructions of infinite families of crossing-critical graphs made essential use ofmultiple edges [11]. On the other hand, constructions such the ones given by Kochol [9],Hlinˇen´y [7], and Bokal [1] deal exclusively with simple graphs.We ask to what extent crossing-criticality is an inherent structural property of a graph,and to what extent crossing-criticality can be induced by multiplying the edges of a (noncrit-ical) graph. Let G, H be graphs. We say that G is obtained by multiplying edges of H if H is a subgraph of G and, for every edge of G , there is an edge of H with the same endvertices. Question 1.
When can a graph be made crossing-critical by multiplying edges? That is, givena (noncritical) graph H , when does there exist a crossing-critical graph G that is obtained bymultiplying edges of H ? Our universe of interest is, of course, the set of nonplanar graphs, since a planar graphobviously remains planar after multiplying any or all of its edges.We show that a large, interesting family of nonplanar graphs satisfy the property inQuestion 1. A nonplanar graph G is near-planar if it has an edge e such that G − e is planar.Near-planar graphs constitute a natural family of nonplanar graphs. Any thought to the ef-fect that crossing number problems might become easy when restricted to near-planar graphsis put definitely to rest by the recent proof by Cabello and Mohar that CrossingNumber is NP-Hard for near-planar graphs [2].Following Geelen et al. [6] (who define internally 3-connectedness for matroids), a graph G is internally -connected if G is simple and 2-connected, and for every separation ( G , G )of G of order two, either | E ( G ) | ≤ | E ( G ) | ≤
2. Hence, internally 3-connected graphsare those that can be obtained from a 3-connected graph by subdividing its edges, with thecondition that no edge can be subdivided more than once.Our main result is that any adequately connected, near-planar graph G obtained byadding an edge to a cubic polyhedral (i.e., planar and 3-connected) graph, belongs to theclass alluded to in Question 1. Theorem 2.
Let G be a near-planar simple graph, with an edge uv such that G − uv is acubic polyhedral graph. Suppose that G − { u, v } is internally -connected. Then there existsa crossing-critical graph that is obtained by multiplying edges of G . We note that some connectivity assumption is needed in order to guarantee that a non-planar graph can be made crossing-critical by multiplying edges. To see this, consider a2raph G which is the 1-sum of a nonplanar graph G plus a planar graph G . Since cross-ing number is additive on the blocks of a graph, it is easy to see that G cannot be madecrossing-critical by multiplying edges.An important ingredient in the proof of Theorem 2 is the following, somewhat curiousstatement for which we could not find any reference in the literature. We recall that a weighted graph is a pair ( G, w ), where G is a graph and w (the weight assignment ) is a mapthat assigns to each edge e of G a number w ( e ), the weight of e . The length of a path in aweighted graph is the sum of the weights of the edges in the path. If u, v are vertices of G ,then the distance d w ( u, v ) from u to v ( under w ) is the length of a minimum length (alsocalled a shortest ) uv -path. The weight assignment w is positive if w ( e ) > e of G , and it is integer if each w ( e ) is an integer. Lemma 3.
Let G be a -connected loopless graph, and let u, v be distinct vertices of G . Thenthere is a positive integer weight assignment such that every edge of G belongs to a shortest uv -path. The rest of this paper is structured as follows.We prove the auxiliary Lemma 3 in Section 2. We then proceed to reformulate Theorem 2in terms of weighted graphs. This simply consists on replacing a multigraph with a weightedsimple graph so that the weights of the edges are the multiplicities of the edges in theoriginal multigraph. As in [3] and [5], this reformulation, carried out in Section 3, turns outto greatly simplify the discussion and the proofs. The equivalent form of Theorem 2, namelyTheorem 4, is then proved in Section 5, the core of this paper. Finally, we present someconcluding remarks and open questions in Section 6.
We use perfect rubber bands, a technique inspired by the work of Tutte [14].Make every edge a perfect rubber band and pin vertices u and v on a board. Since G is 2-connected, and with use of Menger’s theorem, every other vertex w admits two vertex-disjoints paths, one linking w with u and the other linking w with v . Therefore, every vertexwill lie on the segment [ u, v ] on the board. Since Q is dense in R , we may modify verticespositions so that all edge lengths are rational (since they all are barycentric coordinates, itsuffices to have a rational distance between u and v ). We may also modify these coordinateslocally so that no two vertices lie at the same spot.Therefore, every edge lies on a shortest path from u to v .In the end, we may multiply every length by the least common multiple of the denom-inators, so that we get an integer length function on the edges meeting our requirements. (cid:3) Reformulating Theorem 2 in termsof weighted graphs
In the context of Theorem 2, let G be a simple graph which we seek to make crossing-criticalby multiplying (some or all of) its edges. With this in mind, let G be a multigraph (thatis, a graph with multiedges allowed) whose underlying simple graph is G . Now consider the(positive integer) weight assignment w on E ( G ) defined as follows: for each edge uv of G ,let w ( uv ) be the number of edges in G whose endpoints are u and v (i.e., the multiplicity of uv ).If we extend the definition of crossing number to weighted graphs, with the conditionthat a crossing between two edges contributes to the total crossing number by the productsof their weights, then, from the crossing numbers point of view, clearly ( G, w ) captures allthe relevant information from G . In particular, cr( G ) = cr( G, w ). Moreover, by extendingthe definition of crossing-criticality to weighted graphs in the obvious way (which we nowproceed to do), it will follow that G is crossing-critical if and only if ( G, w ) is crossing-critical.To this end, let G be a graph and w a positive integral weight assignment on G . An edge e of ( G, w ) is crossing-critical if cr(
G, w e ) < cr( G, w ), where w e is the weight assignmentdefined by w e ( f ) = w ( f ) for f = e and w e ( e ) = w ( e ) −
1. As with ordinary graphs, (
G, w )is crossing-critical if all its edges are crossing-critical.Under this definition of crossing-criticality for weighted graphs, it is now obvious that ifwe start with a multigraph G and derive its associated weighted graph ( G, w ) as above, then G is crossing-critical if and only if ( G, w ) is crossing-critical.In view of this equivalence (for crossing number purposes) between multigraphs andweighted graphs, it follows that Theorem 2 is equivalent to the following:
Theorem 4 ( Equivalent to Theorem 2).
Let G be a near-planar simple graph, with anedge uv such that G − uv is a cubic polyhedral graph. Suppose that G − { u, v } is internally -connected. Then there exists a positive integer weight assignment w such that ( G, w ) iscrossing-critical. For the rest of this paper: • we let G u,v := G − { u, v } ; and • we refer to the hypotheses that G − uv is 3-connected and G u,v is internally 3-connectedsimply as the connectivity assumptions on G − uv and G u,v , respectively. G u,v and its dual G ∗ u,v Before moving on to the proof of Theorem 5, we establish some facts on the graph G u,v .4 .1 Remarks on the vertices incident with u and v Let u , u , and u be the vertices of G (other than v ) adjacent to u . Analogously, Let v , v , and v be the vertices of G (other than u ) adjacent to v .We start by noting that u , u , u , v , v , v are all distinct. First of all, if i = j , then since G is simple it follows that u i = u j . Now suppose that u i = v j for some i, j , and consider anembedding of G − uv in the plane. It is easy to see that since u i = v j , and G − uv is cubic, itfollows that uv can be added to the embedding of G − uv without introducing any crossings,resulting in an embedding of G . This contradicts the nonplanarity of G . Thus u i = v j forall i, j ∈ { , , } , completing the proof that u , u , u , v , v , v are all distinct. G − uv and G u,v We note that the connectivity assumptions on G − uv and G u,v imply that these two graphsadmit unique (up to homeomorphism) embeddings in the plane. This allows us, for the restof the proof, to regard these as graphs embedded in the plane.Since G u,v is a subgraph of G − uv , it follows that we may assume that the restriction ofthe embedding of G − uv to G u,v is precisely the embedding of G u,v . Conversely, to obtain theembedding of G − uv , we may start with the embedding of G u,v ; then we find the (unique)face F u incident with u , u , and u , and draw uu , uu , and uu (and, of course, u ) inside F u ; and similarly find the (unique) face F v incident with v , v , and v , and draw vv , vv ,and vv (and, of course, v ) inside F v . Note that F u = F v , as otherwise the edge uv couldbe added to the embedding of G − uv without introducing any crossings, resulting in anembedding of G , contradicting its nonplanarity. G ∗ u,v of G u,v We shall make extensive use of weight assignments on the dual (embedded graph) G ∗ u,v of G u,v . We start by noting that G ∗ u,v is well-defined (and admits a unique plane embedding)since G u,v admits a unique plane embedding. As with G − uv and G u,v , this allows us tounambiguously regard G ∗ u,v for the rest of the proof as an embedded graph. We shall let F denote the set of all faces in G u,v (equivalently, the set of all vertices of G ∗ u,v ).A weight assignment λ on G u,v naturally induces a weight assignment λ ∗ on G ∗ u,v , and viceversa: if e is an edge of G u,v and e ∗ is its dual edge in G ∗ u,v , then we simply let λ ∗ ( e ∗ ) = λ ( e ).Trivially, a weight assignment λ on the whole graph G also naturally induces a weightassignment λ ∗ on G ∗ u,v : it suffices to consider the restriction λ of λ to G u,v , and from this weobtain λ ∗ as we just described.If λ ∗ is a weight assignment on G ∗ u,v , then for F, F ′ ∈ F we let d λ ∗ ( F, F ′ ) denote thelength of a shortest F F ′ -path in G ∗ u,v under λ ∗ . We call d λ ∗ ( F, F ′ ) the distance between F and F ′ under λ ∗ .Now since for i = 1 , , u i has degree 2 in G u,v , it follows that u i is in-cident with exactly two faces in G ∗ u,v , one of which is F u ; let F u i denote the other face.5hus it makes sense to define the distance d λ ∗ ( u i , F ) between u i and any face F ∈ F asmin { d λ ∗ ( F u , F ) , d λ ∗ ( F u i , F ) } . We define F v i and d λ ∗ ( v i , F ) analogously, for i = 1 , , G u,v ensures that F u , F u and F u are pairwise distinct.Similarly, F v , F v and F v are pairwise distinct. Note that maybe F u i = F v for some i ∈{ , , } , or F v j = F u for some j ∈ { , , } .Finally, we say that a weight assignment λ ∗ on G ∗ u,v is balanced if each edge e ∗ of G ∗ u,v belongs to a shortest F u F v -path in ( G ∗ u,v , λ ∗ ). First we show (Proposition 5) that if there exists a weight assignment ω on G with certainproperties, then ( G, ω ) is crossing-critical. The existence of a weight assignment with theseproperties is established in Proposition 6, and so Theorem 4 immediately follows.
Proposition 5.
Suppose that ω is a positive integer weight assignment on G with the fol-lowing properties:(1) The induced weight assignment ω ∗ on G ∗ u,v is balanced.(2) For every pair of edges e, e ′ of G u,v , ω ( e ) ω ( e ′ ) > d ω ∗ ( F u , F v ) · ω ( uv ) .(3) d ω ∗ ( u , F ) · ω ( uu ) + d ω ∗ ( u , F ) · ω ( uu ) + d ω ∗ ( u , F ) · ω ( uu ) ≥ d ω ∗ ( F u , F ) · ω ( uv ) , forevery F ∈ F .(4) For each i = 1 , , , there is a face U i ∈ F such that d ω ∗ ( u i , U i ) > and d ω ∗ ( u , U i ) · ω ( uu ) + d ω ∗ ( u , U i ) · ω ( uu ) + d ω ∗ ( u , U i ) · ω ( uu ) = d ω ∗ ( F u , U i ) · ω ( uv ) .(5) d ω ∗ ( v , F ) · ω ( vv ) + d ω ∗ ( v , F ) · ω ( vv ) + d ω ∗ ( v , F ) · ω ( vv ) ≥ d ω ∗ ( F v , F ) · ω ( uv ) , forevery F ∈ F .(6) For each i = 1 , , , there is a face V i ∈ F such that d ω ∗ ( v i , V i ) > and d ω ∗ ( v , V i ) · ω ( vv ) + d ω ∗ ( v , V i ) · ω ( vv ) + d ω ∗ ( v , V i ) · ω ( vv ) = d ω ∗ ( F v , V i ) · ω ( uv ) .(7) For all i, j ∈ { , , } , ω ( uu i ) · ω ( vv j ) < (1 /
9) min { ω ( e ) | e ∈ E ( G u,v ) } . Then ( G, ω ) is crossing-critical.Proof. Throughout the proof, for brevity we let t := ω ( uv ).To help comprehension, we break the proof into several steps.(A) cr( G, ω ) ≤ t · d ω ∗ ( F u , F v ) . Start with the (unique) embedding of G − uv , and draw uv following a shortest F u F v -pathin ( G ∗ u,v , ω ∗ ). Then the sum of the weights of the edges crossed by uv equals the total weightof the shortest F u F v -path, that is, d ω ∗ ( F u , F v ) (here we use the elementary, easy to checkfact that crossings between adjacent edges can always be avoided; in this case, we may draw6 v so that it crosses no edge adjacent to u or v ). Since ω ( uv ) = t , it follows that such adrawing of ( G, ω ) has exactly t · d ω ∗ ( F u , F v ). crossings.(B) cr( G, ω ) = t · d ω ∗ ( F u , F v ) . Consider a crossing-minimal drawing D of ( G, ω ). An immediate consequence of (2) and(A) is that the drawing of G u,v induced by D is an embedding (that is, no two edges of G u,v cross each other in D ).Now let F ′ (respectively, F ′′ ) denote the face of G u,v in which u (respectively, v ) isdrawn in D . Clearly, for i = 1 , , uu i contributes in at least ω ( uu i ) · d ω ∗ ( u i , F ′ )crossings. Analogously, for i = 1 , , vv i contributes in at least ω ( vv i ) · d ω ∗ ( v i , F ′′ )crossings. Thus it follows from (3) and (5) that the edges in { uu , uu , uu , vv , vv , vv } contribute in at least t · d ω ∗ ( F u , F ′ ) + t · d ω ∗ ( F v , F ′′ ) = t · ( d ω ∗ ( F u , F ′ ) + d ω ∗ ( F v , F ′′ )) crossings.On the other hand, since the ends u, v of uv are in faces F ′ and F ′′ , it follows that edge uv contributes in at least t · d ω ∗ ( F ′ , F ′′ ) crossings. We conclude that D has at least t · (cid:0) d ω ∗ ( F u , F ′ ) + d ω ∗ ( F v , F ′′ ) + d ω ∗ ( F ′ , F ′′ ) (cid:1) . Elementary triangle inequality arguments showthat d ω ∗ ( F u , F ′ )+ d ω ∗ ( F v , F ′′ )+ d ω ∗ ( F ′ , F ′′ ) ≥ d ω ∗ ( F u , F v ), and so D has at least t · d ω ∗ ( F u , F v )crossings. Thus cr( G, ω ) ≥ t · d ω ∗ ( F u , F v ). The reverse inequality is given in (A), and so (B)follows.(C) Crossing-criticality of the edges in G u,v and of the edge uv . Let e be any edge in G u,v . We proceed similarly as in (A). Start with the (unique)embedding of G − uv , and draw uv following a shortest F u F v -path in ( G ∗ u,v , ω ∗ ) that includes e ∗ (the existence of such a path is guaranteed by the balancedness of ω ∗ ). This yields adrawing of ( G, ω ) with exactly t · d ω ∗ ( F u , F v ) crossings, in which e and uv cross each other.Since cr( G, ω ) = t · d ω ∗ ( F u , F v ), it follows that e and uv are both crossed in a crossing-minimaldrawing of ( G, ω ). Therefore both e and uv are crossing-critical in ( G, ω ).(D)
Crossing-criticality of the edges uu , uu , uu , vv , vv , and vv . We prove the criticality of uu ; the proof of the criticality of the other edges is totallyanalogous.Consider the (unique) embedding of G u,v . Put u in face U i (see property (4)) and v in face F v . Then draw uu j , for j = 1 , ,
3, adding ω ( uu j ) · d ω ∗ ( u j , U ) crossings with the edges in G u,v .Since crossings between adjacent edges can always be avoided, it follows that uu , uu , uu getdrawn by adding ω ( uu ) · d ω ∗ ( u , U )+ ω ( uu ) · d ω ∗ ( u , U )+ ω ( uu ) · d ω ∗ ( u , U ) = t · d ω ∗ ( F u , U )crossings (using (4)). Finally we draw vv , vv , vv in face F v . Now this last step may addcrossings, but only of the edges vv , vv , vv with the edges uu , uu , uu . In view of (7), thelast step added fewer than 9 · (1 /
9) min { ω ( e ) | e ∈ E ( G u,v ) } = min { ω ( e ) | e ∈ E ( G u,v ) } crossings. We finally draw uv ; since u is in face U and v is in face F v , it follows that uv canbe drawn by adding t · d ω ∗ ( U , F v ) crossings.The described drawing D of G has then fewer than t · d ω ∗ ( F u , U ) + t · d ω ∗ ( U , F v ) +min { ω ( e ) | e ∈ E ( G u,v ) } = t · d ω ∗ ( F u , F v ) + min { ω ( e ) | e ∈ E ( G u,v ) } = cr( G, ω ) +7in { ω ( e ) | e ∈ E ( G u,v ) } crossings, where for the first equality we used the balancednessof ω ∗ , and for the second equality we used (B). Thus cr( D ) < cr( G, ω ) + min { ω ( e ) | e ∈ E ( G u,v ) } .In D , the edge uu contributes in ω ( uu ) · d ω ∗ ( u , U ) crossings; note that (4) implies that ω ( uu ) · d ω ∗ ( u , U ) >
0. Since obviously d ω ∗ ( u , U i ) ≥ min { ω ( e ) | e ∈ E ( G u,v ) } , it followsthat uu contributes in at least min { ω ( e ) | e ∈ E ( G u,v ) } crossings. Thus, if we remove uu we obtain a drawing of G − uu with fewer than cr( G, ω ) crossings. Therefore uu is criticalin ( G, ω ), as claimed.
Proposition 6.
There exists a positive integer weight assignment ω on G that satisfies(1)–(7) in Proposition 5Proof. We start with a balanced positive integer weight assignment µ ∗ on G ∗ u,v . The existenceof such a µ ∗ is guaranteed from Lemma 3, which applies since the connectivity assumptionon G u,v implies that G ∗ u,v is also 3-connected. Let µ denote the (positive integer) weightassignment naturally induced on G u,v . Claim I.
There exists a rational point ( x , x , x ) such that, for every F ∈ F \ { F u } , d µ ∗ ( u , F ) x + d µ ∗ ( u , F ) x + d µ ∗ ( u , F ) x ≥ d µ ∗ ( F u , F ) . Moreover, there exist faces U , U , U of G u,v , such that d µ ∗ ( u i , U i ) > and, for i = 1 , , , d µ ∗ ( u , U i ) x + d µ ∗ ( u , U i ) x + d µ ∗ ( u , U i ) x = d µ ∗ ( F u , U i ) . Proof.
Let I be the system of inequalities I := { d µ ∗ ( u , F ) x + d µ ∗ ( u , F ) x + d µ ∗ ( u , F ) x ≥ d µ ∗ ( F u , F ) | f ∈ F \ { F u }} , and let Λ denote the set of all triples ( x , x , x ) that satisfy allinequalities in I . Clearly, if x , x , and x are all large enough then ( x , x , x ) is in Λ. ThusΛ is a nonempty convex polyhedron in R .Each inequality J in I naturally defines a plane in R , namely the plane obtained bysubstituting ≥ with =. We call this the plane associated to J .Since each u i is incident with face F u i , it follows that d µ ∗ ( u , F u ) = d µ ∗ ( u , F u ) = d µ ∗ ( u , F u ) = 0, Therefore the inequalities corresponding to F u , F u and F u , respectively,define the following system Γ: d µ ∗ ( u , F u ) x + d µ ∗ ( u , F u ) x ≥ d µ ∗ ( F u , F u ) (Γ1) d µ ∗ ( u , F u ) x + d µ ∗ ( u , F u ) x ≥ d µ ∗ ( F u , F u ) (Γ2) d µ ∗ ( u , F u ) x + d µ ∗ ( u , F u ) x ≥ d µ ∗ ( F u , F u ) (Γ3)where all coefficients (since the faces F u , F u , F u are pairwise distinct) and all the right-hand sides are strictly positive integers. We refer to this as the positive integrality propertyof (Γ1), (Γ2), and (Γ3).This positive integrality property implies that the planes associated to (Γ1) and (Γ2)intersect in a line whose intersection with the positive octant is a (full one-dimensional)8egment. In particular, there is a point ( a , a , a ), all of whose points are positive andrational, and that lies on the intersection of the planes associated to (Γ1) and (Γ2).Now consider the set of points r → := { ( x , a , a ) | x ≥ } . Thus r → is a ray starting at(0 , a , a ) and parallel to the x -axis, lying on the plane associated to (Γ1). Now since (Γ1)is the only inequality in I whose x coefficient is zero, it follows that for every large enough x , the point ( x , a , a ) is in Λ. Now every inequality in I distinct from (Γ1) intersects r → in at most one point. Let a ′ be largest possible such that ( a ′ , a , a ) is the intersection of r → with a plane associated to an inequality I in I \ { (Γ1) } ; since a is the intersection of(Γ2) with (Γ1), it follows that a ′ is well-defined.Since a ′ , a , a all arise from the intersection of planes with integer coefficients, it followsthat they are all rational numbers. We claim that the rational point ( a ′ , a , a ) satisfies therequirements in the Claim.Consider any inequality J in I distinct from (Γ1), (Γ2), and (Γ3). Thus all the coefficientsof J are nonzero, and so the intersection of the plane associated to J with the positive octantis a triangle. Let ( j, a , a ) be the intersection of this triangle with the line { ( x , a , a ) | x ∈ R } . Then j ≤ a ′ , and so the ray { ( x , a , a ) | x ∈ R , x ≥ j } is contained in the feasible region of J (that is, the region of R that consists of those points for which J is satisfied).In particular, ( a ′ , a , a ) is in the feasible region of J . A similar argument shows that alsoin the case in which J is either (Γ2) or (Γ3), then ( a ′ , a , a ) is in the feasible region of J .This proves the first part of Claim I.Now we recall that ( a ′ , a , a ) lies on the plane associated to (Γ1), that is, d µ ∗ ( u , F u ) a + d µ ∗ ( u , F u ) a = d µ ∗ ( F u , F u ). Since d µ ∗ ( u , F u ) = 0, we have d µ ∗ ( u , F u ) a ′ + d µ ∗ ( u , F u ) a + d µ ∗ ( u , F u ) a = d µ ∗ ( F u , F u ). Noting that d µ ∗ ( u , F u ) > d µ ∗ ( u e , F u ) >
0, the second part of Claim I follows for i = 2 and 3 by setting U = U = F u .Now let U be the face in G associated to inequality I . Thus d µ ∗ ( u , U ) a ′ + d µ ∗ ( u , U ) a + d µ ∗ ( u , U ) a = d µ ∗ ( F u , U ). We recall that (Γ1) is the only inequality in I whose x coeffi-cient is 0; since inequality I is distinct from (Γ1) it follows that d µ ∗ ( u , U ) >
0. Thus thesecond part of Claim I follows for i = 1 for this choice of U .The proof of the following statement is totally analogous: Claim II.
There exists a rational point ( y , y , y ) such that, for every F ∈ F \ { F v } , d µ ∗ ( v , F ) y + d µ ∗ ( v , F ) y + d µ ∗ ( v , F ) y ≥ d µ ∗ ( F v , F ) . Moreover, there exist faces V , V , V of G u,v , such that d µ ∗ ( v i , V i ) > and, for i = 1 , , , d µ ∗ ( v , V i ) y + d µ ∗ ( v , V i ) y + d µ ∗ ( v , V i ) y = d µ ∗ ( F u , V i ) . (cid:3) Let ( p /q , p /q , p /q ) be a point as in Claim I, and let ( a /b , a /b , a /b ) be a point asin Claim II, where all p i s, q i s, a i s, and b i s are integers. Let M := q q q b b b , and let r := p q q b b b , r := p q q b b b , r := p q q b b b , s := a b b q q q , s := a b b q q q , and s := a b b q q q . 9hen ( r , r , r ) is a positive integer solution to the set of inequalities { d µ ∗ ( u , F ) r + d µ ∗ ( u , F ) r + d µ ∗ ( u , F ) r ≥ M · d µ ∗ ( F u , F ) : F ∈ F \ { F u }} , and for each i = 1 , ,
3, wehave d µ ∗ ( u , U i ) r + d µ ∗ ( u , U i ) r + d µ ∗ ( u , U i ) r = M · d µ ∗ ( F u , U i ).Similarly, ( s , s , s ) is a positive integer solution to the set of inequalities { d µ ∗ ( v , F ) s + d µ ∗ ( v , F ) s + d µ ∗ ( v , F ) s ≥ M · d µ ∗ ( F v , F ) : F ∈ F \ { F v }} , and for each i = 1 , ,
3, wehave d µ ∗ ( v , V i ) s + d µ ∗ ( v , V i ) s + d µ ∗ ( v , V i ) s = M · d µ ∗ ( F v , V i ).Finally, let c be any integer greater than M · d µ ∗ ( F u , F v ) / (min { µ ( e ) | e ∈ E ( G u,v ) } ) andalso greater than 9 r i s j / min { µ ( e ) | e ∈ E ( G u,v ) } , for all i, j ∈ { , , } .Define the weight assignment ω on G as follows: • ω ( uv ) = M ; • ω ( uu i ) = r i and ω ( vv i ) = s i for i = 1 , , • ω ( e ) = c · µ ( e ), for all edges e in G u,v .We claim that ω (and its induced weight assignment ω ∗ on G ∗ u,v ) satisfies (1)–(7) inProposition 5.To see that ω ∗ satisfies (1), it suffices to note that ω ∗ inherits the balancedness (whenrestricted to G ∗ u,v ) from µ ∗ .Now let e, e ′ be edges of G u,v . Then ω ( e ) ω ( e ′ ) = c · µ ( e ) µ ( e ′ ) ≥ c · (min { µ ( f ) | f ∈ E ( G u,v ) } ) > c · M · d µ ∗ ( F u , F v ) = ω ( uv )( c · d µ ∗ ( F u , F v )) = ω ( uv ) · d ω ∗ ( F u , F v ). This proves(2).For (3) and (4), recall that ( r , r , r ) is a positive integer solution to the set of inequalities { d µ ∗ ( u , F ) r + d µ ∗ ( u , F ) r + d µ ∗ ( u , F ) r ≥ M · d µ ∗ ( F u , F ) : F ∈ F \ { F u }} , and for each i = 1 , ,
3, we have d µ ∗ ( u , U i ) r + d µ ∗ ( u , U i ) r + d µ ∗ ( u , U i ) r = M · d µ ∗ ( F u , U i ). Thedefinition of ω (and its induced ω ∗ ) then immediately imply (3) and (4) (we are using thatfor any faces F, F ′ ∈ F \ { F u } , d ω ∗ ( F, F ′ ) = c · d µ ∗ ( F, F ′ )). The proof that (5) and (6) holdis totally analogous.Finally, we recall that we defined c so that c > r i s j / min { µ ( e ) | e ∈ E ( G u,v ) } for all i, j ∈ { , , } . By the definition of ω , this is equivalent to c · min { µ ( e ) | e ∈ E ( G u,v ) } > ω ( uu i ) ω ( vv j ), that is, min { ω ( e ) | e ∈ E ( G u,v ) } > ω ( uu i ) ω ( vv j ), which is in turn obviouslyequivalent to (7). Let G be the class of graphs that can be made crossing-critical by a suitable multiplication ofedges. In this work we have proved that a large family of graphs is contained in G (note thatthe cubic condition is only used around vertices u, v, u , u , u , v , v and v ; other verticescan have arbitrary degrees). Which other graphs belong to G ? Is there any hope of fullycharacterizing G ?It is not difficult to prove that we can restrict our attention to simple graphs: if G is agraph with multiple edges and G is a maximal simple graph contained in G , then G is in G if and only if G is in G . 10es´us Lea˜nos has observed that the graph K +3 , obtained by adding to K , an edge(between vertices in the same chromatic class) is not in G . Following ˇSir´aˇn [12, 13], an edge e in a graph G is a Kuratowski edge if there is a subgraph H of G that contains e and ishomeomorphic to a Kuratowski graph (that is, K , or K ). It is trivial to see that the addededge in Lea˜nos’s example is not a Kuratowski edge of K +3 , . This observation naturally givesrise to the following. Conjecture 7. If G is a graph all whose edges are Kuratowski edges, then G can be madecrossing-critical by a suitable multiplication of its edges. We remark that the converse of this statement is not true: ˇSir´aˇn [12] gave examples ofgraphs that contain crossing-critical edges that are not Kuratowski edges.The only positive result we have in this direction is that Kuratowski edges can be madeindividually crossing-critical:
Proposition 8. If e is a Kuratowski edge of a graph G , then e can be made crossing-criticalby a suitable multiplication of the edges of G .Proof. Let H be a subgraph of G , homeomorphic to a Kuratowski graph, such that e is in H . Let f be another edge of H such that there is a drawing D H of H with exactly onecrossing, which involves e and f . Extend D H to a drawing D of G . Let p be the number ofcrossings in D . If p = 1 then e is already critical in G , so there is nothing to prove. Thuswe may assume that p ≥
2. Add p − e and f , add p − H \ { e, f } , and do not add any parallel edge to the other edges of G .Let G ′ denote the resulting graph.We claim that cr( G ′ ) ≤ p . To see this, consider the drawing D ′ of G ′ naturally inducedby D . It is easy to check that each crossing from D yields at most p crossings in D ′ (herewe use that e and f are the only edges in H that cross each other in D ). Thus D ′ has atmost p · p = p crossings, and so cr( G ′ ) ≤ p , as claimed.On the other hand, it is clear that a drawing of G ′ in which e and f do not cross eachother has at least p crossings. Since p > p ≥ cr( G ′ ), it follows that no such drawingcan be optimal. Therefore e and f cross each other in every optimal drawing of G ′ . Thisimmediately implies that e is critical in G ′ .The immediate next step towards Conjecture 7 seems already difficult enough so as toprompt us to state it: Conjecture 9.
Suppose that e, f are Kuratowski edges of a graph G . Then there exists agraph H , obtained by multiplying edges of G , such that both e and f are crossing-critical in H . Acknowledgements
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