Improvement on the Decay of Crossing Numbers

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1 Grahs and Combinatorics 2013) 29: DOI /s ORIGINAL PAPER Imrovement on the Decay of Crossing Numbers Jakub Černý Jan Kynčl Géza Tóth Received: 24 Aril 2007 / Revised: 1 November 2011 / Published online: 25 Febuary 2012 Sringer 2012 Abstract We rove that the crossing number of a grah decays in a continuous fashion in the following sense. For any ε > 0 there is a δ > 0 such that for a sufficiently large n, every grah G with n vertices and m n 1+ε edges, has a subgrah G of at most 1 δ)m edges and crossing number at least 1 ε)crg). This generalizes the result of J. Fox and Cs. Tóth. Keywords Crossing number Embedding method 1 Introduction For any grah G, let ng) res. mg)) denote the number of its vertices res. edges). If it is clear from the context, we simly write n and m instead of ng) and mg). The crossing number crg) of a grah G is the minimum number of edge crossings over all drawings of G in the lane. In the otimal drawing of G, crossings are not necessarily The first and the second author were suorted by Program Structuring the Euroean Research Area, Grant MRTN-CT at Rényi Institute and by roject CE-ITI GACR P2020/12/G061) of the Czech Science Foundation. The second author was also artially suorted by GraDR EUROGIGA roject GIG/11/E023 and by the grant SVV Discrete Methods and Algorithms). The third author was suorted by the Hungarian Research Fund grants OTKA T , T , K-60427, and K A reliminary version of this aer aeared in the roceedings of Grah Drawing 2007 [2]. J. Černý J. Kynčl Deartment of Alied Mathematics and Institute for Theoretical Comuter Science, Faculty of Mathematics and Physics, Charles University, Prague, Czech Reublic kuba@kam.mff.cuni.cz J. Kynčl kyncl@kam.mff.cuni.cz G. Tóth B) Rényi Institute, Hungarian Academy of Sciences, Budaest, Hungary geza@renyi.hu

2 366 Grahs and Combinatorics 2013) 29: distributed uniformly among the edges. Some edges could be more resonsible for the crossing number than some other edges. For any fixed k, it is not hard to construct a grah G whose crossing number is k, but G has an edge e such that G \ e is lanar. Richter and Thomassen [7] started to investigate the following general roblem. We have a grah G, and we want to remove a given number of edges. By at least how much does the crossing number decrease? They conjectured that there is a constant c such that every grah G with crg) = k has an edge e with crg e) k c k. They only roved that G has an edge with crg e) 2 5 k O1). Pach et al. [5] roved that for every grah G with mg) ng), we have crg) m3 n 2. It is not hard to see [6] that for any edge e, we have crg e) crg) m + 1. These two results imly an imrovement of the Richter Thomassen bound if m 8.1n, and also imly the Richter Thomassen conjecture for grahs of n 2 ) edges. Fox and Tóth [3] investigated the case where we want to delete a ositive fraction of the edges. Theorem A [3] For every ε>0, thereisann ε such that every grah G with ng) n ε vertices and mg) ng) 1+ε edges has a subgrah G with mg ) 1 ε ) mg) 24 and crg ) ) 1 28 o1) crg). In this note we generalize Theorem A. Theorem For every ε, γ > 0, thereisann ε,γ such that every grah G with ng) n ε,γ vertices and mg) ng) 1+ε edges has a subgrah G with and mg ) 1 εγ ) mg) 1224 crg ) 1 γ)crg). 2 Proof of the Theorem Our roof is based on the argument of Fox and Tóth [3], the only new ingredient is Lemma 1. Definition Let r 2, 1 be integers. A 2r-earring of size is a grah which is a union of an edge uv and edge-disjoint aths between u and v, each of length at most 2r 1. Edge uv is called the main edge of the 2r-earring.

3 Grahs and Combinatorics 2013) 29: Lemma 1 Let r 2, 1 be integers. There exists n 0 such that every grah G with n n 0 vertices and m 6rn 1+1/r edges contains at least m/3r edge-disjoint 2r-earrings, each of size. Proof By the result of Alon et al. [1], for some n 0, every grah with n n 0 vertices and at least n 1+1/r edges contains a cycle of length at most 2r. Suose that G has n n 0 vertices and m 6rn 1+1/r edges. Take a maximal edge-disjoint set {E 1, E 2,...,E x } of 2r-earrings, each of size. Let E = E 1 E 2 E x, the set of all edges of the earrings and let G = G E. Now let E 1 bea2rearring of G of maximum size. Note that this size is less than. Let G 1 = G E 1. Similarly, let E 2 be a 2r-earring of G 1 of maximum size and let G 2 = G 1 E 2. Continue analogously, as long as there is a 2r-earring in the remaining grah. We obtain the 2r-earrings E 1, E 2,...,E y, and the remaining grah G = G y does not contain any 2r-earring. Let E = E 1 E 2 E y. We claim that y < n 1+1/r. Suose on the contrary that y n 1+1/r. Take the main edges of E 1, E 2,...,E y. We have at least n1+1/r edges so by the result of Alon et al. [1] some of them form a cycle C of length at most 2r. Let i be the smallest index with the roerty that C contains the main edge of E i. Then C, together with E i would be a2r-earring of G i 1 of greater size than E i, contradicting the maximality of E i. Each of the earrings E 1, E 2,...,E y has at most 1)2r 1) + 1 edges so we have E y 1)2r 1) + y <2r 1)n 1+1/r. The remaining grah, G does not contain any 2r-earring, in articular, it does not contain any cycle of length at most 2r, since it is a 2r-earring of size one. Therefore, by [1], for the number of its edges we have eg )<n 1+1/r. It follows that the set E ={E 1, E 2,...,E x } contains at least m 2rn 1+1/r 2 3 m edges. Each of E 1, E 2,...,E x has at most 2r 1) + 1 2r edges, therefore, x m/3r. Lemma 2 [3] Let G be a grah with n vertices, m edges, and degree sequence d 1 d 2 d n. Let l be the integer such that l 1 i=1 d i < 4m/3 but l i=1 d i 4m/3. If n is large enough and m = n log 2 n) then crg) 1 65 l di 2. Proof of the Theorem Let ε, γ 0, 1) be fixed. Choose integers r, such that 1 ε < r 2 67 ε, and γ < 68 γ. It follows that we have r 1 <ε r 2, and 67 <γ 68. Then there is an n ε,γ with the following roerties: a) n ε,γ n 0 from Lemma 1,b) n ε,γ ) 1+ε > 18r n ε,γ ) 1+1/r. Let G be a grah with n n ε,γ vertices and m n 1+ε edges. Let v 1,...,v n be the vertices of G, of degrees d 1 d 2 d n and define l as in Lemma 2, that is, l 1 i=1 d i < 4m/3 but l i=1 d i 4m/3. Let G 0 be the subgrah of G induced by v 1,...,v l. Observe that G 0 has m m/3 edges. Therefore, by Lemma 1 G 0 contains at least m /3r m/9r edge-disjoint 2r-earrings, each of size. Let M be the set of the main edges of these 2r-earrings. We have M m/9r εγ 1224 m. Let G = G M and G 0 = G 0 M. i=1

4 368 Grahs and Combinatorics 2013) 29: v i a) Fig. 1 The thick edges are edges of G, the thin edges are the otential edges. a A neighborhood of a crossing in DG ) and b a neighborhood of a vertex v i in G b) Take an otimal drawing DG ) of the subgrah G G. We have to draw the missing edges to obtain a drawing of G. Our method is a randomized variation of the embedding method, which has been alied by Leighton [4], Richter and Thomassen [7], Shahrokhi et al. [8], Székely [9], and most recently by Fox and Tóth [3]. For every missing edge e i = u i v i M G 0, e i is the deleted main edge of a 2r-earring E i G 0. So there are edge-disjoint aths in G 0 from u i to v i. For each of these aths, draw a curve from u i to v i infinitesimally close to that ath, on either side. Call these curves otential u i v i -edges and call the resulting drawing D. Note that a otential u i v i -edge crosses itself if the corresonding ath does. In such cases, we redraw the otential u i v i -edge in the neighborhood of each self-crossing to get a noncrossing curve. To get a drawing of G, for each e i = u i v i M, choose one of the otential u i v i -edges at random, indeendently and uniformly, with robability 1/, and draw the edge u i v i as that curve. There are two tyes of new crossings in the obtained drawing of G. First category crossings are infinitesimally close to a crossing in DG ), second category crossings are infinitesimally close to a vertex of G 0 in DG ). The exected number of first category crossings is at most ) 2 crg ) = ) 2 crg ). Indeed, for each edge of G, there can be at most one new edge drawn next to it, and that is drawn with robability at most 1/. Therefore, in the close neighborhood of a crossing in DG ), the exected number of crossings is at most ) see 2 Fig. 1a). In order to estimate the exected number of second category crossings, consider the drawing D near a vertex v i of G 0. In the neighborhood of vertex v i we have at most d i original edges. Since we draw at most one otential edge along each original edge, there can be at most d i otential edges in the neighborhood. Each otential edge can cross each original edge at most once, and any two otential edges can cross at most twice see Fig. 1b). Therefore, the total number of first category crossings in D in the neighborhood of v i is at most 2di 2. This bound can be substantially imroved

5 Grahs and Combinatorics 2013) 29: with a more careful argument, see e. g. [3], but we do not need anything better here.) To obtain the drawing of G, we kee each of the otential edges with robability 1/, so the exected number of crossings in the neighborhood of v i is at most )d 2 i, using the fact that the self-crossings of the otential uv-edges have been eliminated. Therefore, the total exected number of crossings in the random drawing of G is at most )crg ) ) l i=1 d 2 i. There exists an embedding with at most this many crossings, therefore, by Lemma 2 we have crg) ) 2 1 crg ) ) l 2 i= ) 2 crg ) d 2 i ) crg). It follows that ) 2 crg) ) 2 crg ), so ) ) 2 crg) 1 1 ) 2 2 crg ), ) ) crg) crg ), 1 67 ) crg) crg ), consequently, 1 γ ) crg) crg ). 3 Concluding Remarks In the statement of our Theorem we cannot require that every subgrah G with 1 δ) mg) edges has crossing number crg ) 1 γ)crg), instead of just one such subgrah G. In fact, the following statement holds. Proosition 1 For every ε 0, 1) there exist grahs G n with ng n ) = Θn) vertices and mg n ) = Θn 1+ε ) edges with subgrahs G n G n such that mg n ) = 1 o1)) mg n)

6 370 Grahs and Combinatorics 2013) 29: and crg n ) = ocrg n)). Proof Roughly seaking, G n will be the disjoint union of a large grah G n with low crossing number and a small grah H n with large crossing number. More recisely, let G = G n be a disjoint union of grahs G = G n and H = H n, where G is a disjoint union of Θn 1 ε ) comlete grahs, each with n ε vertices and H is a comlete grah with n 3+5ε)/8 vertices. We have mg) = Θn 1+ε ) and mh) = Θn 3+5ε)/4 ) = omg)), since 3+5ε 4 < 1 + ε. By the crossing lemma see e. g. [5]), crg) crh) = n 3+5ε)/2 ), but crg ) = On 1 ε n 4ε ) = On 1+3ε ) = ocrg)), because 3+5ε 2 > 1 + 3ε. In the reliminary version of this aer [2] we conjectured that we can require that a ositive fraction of all subgrahs G of G with 1 δ)mg) edges has crossing number crg ) 1 γ)crg). The following construction shows that the conjecture does not hold in general for grahs with less than n 4/3 1) edges. Proosition 2 For every ε 0, 1/3) and δ>0 there exist grahs G n with ng n ) = Θn) vertices and mg n ) = Θn 1+ε ) edges with the following roerty. Let G n be a random subgrah of G n such that we choose each edge of G n indeendently with robability = 1 δ. Then Pr [ crg n ) ocrg n)) ] > 1 e δnθ1/3 ε). Proof As in Proosition 1, the idea is to build the grah G = G n from two disjoint grahs K and H, where K is a large grah with low crossing number and H is a small grah with high crossing number. In addition, deleting a random constant fraction of edges from H will break all the crossings in H with high robability. Now we describe the constructions more recisely. Let γ>0 be a constant such that 3ε + 4γ < 1. Let K be a disjoint union of Θn 1 ε ) comlete grahs, each with n ε vertices we omit the exlicit rounding to kee the notation simle). We have mk ) = Θn 1+ε ) and crk ) = Θn 1+3ε ). The grah H consists of five main vertices v 1,v 2,...,v 5 and n 1 2γ internally vertex disjoint aths of length n γ connecting each air v i,v j. The grah H has nh) = Θn 1 γ ) vertices and mh) = Θn 1 γ ) edges. We claim that crh) = n 2 4γ. The uer bound follows from the fact that the crossing number of K 5 is 1. We take a drawing of K 5 with one crossing and relace each edge e by n 1 2γ aths drawn close to e. For the lower bound take a drawing of H with minimum number of crossings. Let i, j be a ath with the minimum number of crossings among the aths connecting v i and v j. By redrawing all the other aths connecting v i and v j along i, j the crossing number of the drawing does not change. The aths i, j together form a subdivision of K 5, therefore at least one air i, j, k,l of the aths crosses. Due to the redrawing, every ath connecting v i and v j crosses every ath connecting v k and v l, which makes n 2 4γ crossings. By the choice of γ,n 1+3ε = on 2 4γ ), therefore crg) = ΘcrH)) and crk ) = ocrg)).

7 Grahs and Combinatorics 2013) 29: Let G be a random subgrah of G where each edge of G is taken indeendently with robability = 1 δ. Let H = G H. We show that with high robability, H is a forest, in articular crh ) = 0. This haens if at least one edge is missing from every ath connecting two main vertices of H. The robability of such an event is at least 1 n 1 δ) nγ 1 e δnγ +log n. It follows that with this robability, crg ) crk ) ocrg)). Note that in the above construction the number δ does not have to be constant: it is enough to delete a random δ = c log n/n γ fraction of the edges to get the same conclusion with robability almost 1. The question whether deleting a small random constant fraction of the edges of a grah G decreases the crossing number only by a small constant fraction remains oen for grahs with more than n 4/3 edges. We do not know the answer even to the following weaker version of the question. Problem 1 Let ε 0, 2/3) and 0, 1) be constants. Does there exist c) >0 and n 0 such that for every grah G with ng) >n 0 and mg) >ng) 4/3+ε, a random subgrah G of G with each edge taken with robability has crossing number at least c) crg), with robability at least 1/2? The grahs in Proosition 2 have small number of edges resonsible for almost all the crossings. Is this the only way how to force a random subgrah of G to have crossing number ocrg))? Problem 2 Let ε>0. Does there exist n 0 and δ such that every grah G with ng) n 0 and mg) ng) 1+ε has a subset F of omg)) edges such that every subgrah G of G with mg ) 1 δ)mg) and F EG ) has crg ) 1 ε)crg)? References 1. Alon, N., Hoory, S., Linial, N.: The Moore bound for irregular grahs. Grahs Combin. 18, ) 2. Černý, J., Kynčl, J., Tóth, G.: Imrovement on the decay of crossing numbers. In: Proceedings of the 15th International Symosium on Grah Drawing Grah Drawing 2007). Lecture Notes in Comuter Science, vol. 4875, Sringer, Berlin 2008) 3. Fox, J., Cs, Tóth: On the decay of crossing numbers. J. Combin. Theory Ser. B 981), ) 4. Leighton, T.: Comlexity Issues in VLSI. MIT Press, Cambridge 1983) 5. Pach, J., Radoičić, R., Tardos, G., Tóth, G.: Imroving the Crossing Lemma by finding more crossings in sarse grahs. In: Proceedings of the 19th ACM Symosium on Comutational Geometry, ) [Also in Discrete Comut. Geom. 36, )] 6. Pach, J., Tóth, G.: Thirteen roblems on crossing numbers. Geombinatorics 9, ) 7. Richter, B., Thomassen, C.: Minimal grahs with crossing number at least k. J. Combin. Theory Ser. B 58, ) 8. Shahrokhi, F., Sýkora, O., Székely, L., Vrt o, I.: Crossing numbers: bounds and alications. In: Intuitive Geometry Budaest, 1995), vol. 6, Bolyai Society Mathematical Studies, Budaest 1997) 9. Székely, L.: Short roof for a theorem of Pach, Sencer, and Tóth. In: Towards a Theory of Geometric Grahs. Contemorary Mathematics, vol. 342, AMS, Providence 2004)