Lower bounds for the Crossing Number of the Cartesian Product of a Vertex-transitive Graph with a Cycle

Transcription

1 Lower bounds for the Crossng Number of the Cartesan Product of a Vertex-transtve Graph wth a Cyce Junho Won MIT-PRIMES December 4, 013 Abstract. The mnmum number of crossngs for a drawngs of a gven graph G on a pane s caed ts crossng number, denoted crg. Exact crossng numbers are known ony for a few fames of graphs, and even the crossng number of a compete graph K m s not known for a m. Wenpng et a. showed that crk m C n n crk m+ for n 4 and m 4. We adopt ther method to fnd a ower bound for crg C n where G s a vertex-transtve graph of degree at east 3. We aso suggest some partcuar vertex-transtve graphs of nterest, and gve two coroares that gve ower bounds for crg C n n terms of n, crg, the number of vertces of G, and the degree of G, whch mprove on Wenpng et a. s resut. 1 Introducton For basc defntons and notatons that are not expaned, the readers are referred to Deste [3]. If G s a graph, we denote ts vertex set by V G and ts edge set by EG. C n, or the n-cyce, s the graph wth some n vertces {v 1,..., v n } wth an edge set {v 1 v,..., v v n, v n v 1 }. K n, or the compete graph on n vertces, s the smpe graph wth n vertces n whch any two vertces are oned by an edge. K,m denotes the graph whose vertex set can be parttoned nto two subsets of sze and m, such that any two vertces n the same subset are not oned and any two vertces n dfferent subsets are oned. Cacuatng the crossng number of a gven graph s a maor area of research n topoogca graph theory. It has proven to be a very dffcut task, and there are ony few fames of graphs whose crossng numbers are known. In fact, n 1983 Garey and Johnson [3] showed that the cacuaton s NP-compete. However, crossng numbers of some graphs are known, and one of the most nterestng fames of graphs have been the Cartesan products of two eementary graphs, such as paths, cyces, stars, compete graphs, compete bpartte or mutpartte graphs see, for exampe, Kešč [5]. A maor resut was acheved by Gebsky and Saazar [4] when they cacuated the crossng number of the Cartesan product of two cyces, crc m C n, for a but fntey many n greater than each gven m: Theorem 1.1 Gebsky and Saazar, 004. If n mm + 1 and m 3, then crc m C n = m n. In a smar ne, we contnue nvestgatng the crossng number of Cartesan products of graphs, but n ths case, a much arger famy of graphs: namey, the Cartesan product of any vertex-transtve graph G of degree at east 3 wth a cyce. We obtan ower bounds for ther crossng numbers n terms of a sma graph G, whose order s bgger than that of G. We ony consder fnte smpe undrected graphs. Let G be a graph wth a vertex set V and an edge set E. We ony consder good drawngs of G, n whch 1. no edge crosses tsef,. no ncdent edges cross, 3. no more than two edges cross at a common pont, 4. edges do not cross vertces, and 1

2 5. edges that cross do so ony once. The frst two types of crossngs can aways be emnated, and the next three condtons are by our choce. We denote the crossng number of G for the pane by crg. If DG s a good drawng of G, then we denote by vdg the number of crossngs n DG. The Cartesan product G H of graphs G and H has vertex set V G V H and edge set EG H = {{x 1, y 1, x, y } x 1 = x and y 1 y EH or y 1 = y and x 1 x EG}. In 008, Wenpng et. a [7] showed the foowng: Lemma 1. Wenpng et a., 008. crk m C n n crk m+ for n 4 and m 4. Usng ther method, we prove a much more genera theorem whch appes to any vertex transtve graph, a graph such that for any two vertces v 1, v there exsts a graph automorphsm on the vertex set that maps v 1 to v. Of course, the compete graph K m s vertex transtve. The key observaton s that, for any vertex-transtve graph G, every subgraph formed by the unon of a copy of G and a copy of C n n G C n are somorphc to each other. In the foowng theorem, the graph G s obtaned from G by addng two vertces, fxng any vertex n G that we ca v 0, and onng each of the two new vertces to v 0 and a of ts neghbors as we as to each other. Theorem 1.3 Man Theorem. Suppose that G s a vertex-transtve graph wth degree p 3. Let us denote V G = m. Then for n 4 and m 4, we have the foowng ower bound for the crossng number of the Cartesan product of G and C n : [ ] m crg C n p + 1 m crg 1 crg n. p + 1 Coroary 1.4. In the same condton, crg C n n crg. Observe that Wenpng et a. s resut s a partcuar case of the above coroary. The proof s smpe. Proof. Snce m p+1 and crg crg, snce G s a subgraph of G, the rght-hand sde of ths nequaty s at most that of the nequaty n Theorem 1. They are equa f and ony f G = K m, the compete graph of order m 4, n whch case m = p + 1. It s known that crk 3,n = n see Rchter and Sǐrán, [6]. Observe that for any vertex transtve graph G, the subgraph of G above nduced by v 0, ts adacent vertces, and the two new vertces has K 3,p as a subgraph. Aso, the ntersecton of ths K 3,p and the orgna copy of G n G s somorhc to the star wth p + 1 vertces. Snce a star does not cross tsef n a good drawng, no crossng of the K 3,p n a good drawng occurs wthn the copy of G. Therefore, we have crg crg crk 3,p = p p 1, and obtan the foowng coroary: Coroary 1.5. Suppose that G s a vertex-transtve graph wth degree p 3. Let us denote V G = m. Then for n 4 and m 4, crg C n m p p 1 n + crg n. p + 1 Ths coroary shows that we can aways obtan a ower bound that s better than the obvous one, crg n. Notce that for a random graph G, we have crg m m = Om 4. It s of nterest for future research to fnd a ower bound that s asymptotcay greater than crg n by a constant mutpe, or a ower bound that s Ωm 4, f possbe.

3 Proof of the Man Theorem Proof. Parts of the proof we gve here for Theorem 1.3 cosey foow the countng method deveoped for the proof of Theorem. n Wenpng et a. [8]. However, for competeness and carfcaton of dfferences n the proofs we w ncude the entre proof wth ony a few omssons and aso re-ntroduce some borrowed notatons and emmas. In partcuar, f A, B are two dsont subsets of EG, the number of crossngs between edges n A and edges n B n a drawng D s denoted by v D A, B. The number of crossngs that occur between edges of A s denoted by v D A, so that vd=v D G. Aso, f X s a subset of V G or EG for a gven graph G then G[X] denotes the subgraph of G nduced by X. We borrow the foowng straghtforward emma: Lemma.1 Wenpng et a., 008. Let A, B, C be mutuay dsont subsets of EG. Then, v D C, A B = v D C, A + v D C, B, and v D A B = v D A + v D B + v D A, B. Throughout ths proof, we consder G to be a fnte smpe undrected graph that s vertex-transtve wth a reguar degree p. Let V G = {v 0,..., v } so that V G = m. Let G be defned as n Theorem 1.3 we may aso use the apostrophe notaton for a graph obtaned smary from a vertex-transtve graph other than G. In order to smpfy the notatons, we defne a functon f : J J {0, 1}, where J = {0,..., }, reated to{ the adacency matrx of G as n the foowng: 0, f v x v y / EG fx, y = 1, f v x v y EG. Observe that fx, x = 0 for a x J, and f G = K m for some m, then the vaue of fx, y = 1 ff x y. Now et us consder our Cartesan product G C n = H. Usng f, we can defne the vertex set and edge set of H as n the foowng: V H = {v 0 m 1, 0 n 1}, EH = {v v k f, k = 1} {v 1 v 0 m 1}. Here and throughout Secton, superscrpts are read moduo n and subscrpts are read moduo m. We anayze H by consderng t as a dsont unon of subsets of V H and EH. For 0 n 1, et V = {v 0 m 1}, E = {v v k f, k = 1}, G = V, E, and M = {v 1 v 0 m 1}. Then E E = for 0 < n 1, M M = for 0 < n 1, E M = for 0 n 1, 0 n 1, EH = E M. For any drawng D of H we have v D H = v D E M. By Lemma.1 t foows that v D H = v D E + 0 < v D E, E + v DM + Furthermore, by consderng the party of n, we obtan 0 < v D M, M + v D E, M. v D H = v D E + v D M + + v D E, E + + v D M, M + v D E, M n + 1 mod n + 1 mod v D E, E + n v D M, M + n. 1 3

4 Fgure 1: H and H[E ]. Now, et us consder dfferent subsets of EH. From here on, we fx the range of to be 0 m 1, and that of k to be 0 n 1. Let E = {v v f, = 1, M = {v 1 v = or f, = 1}, R = {v 0v1, v1 v,..., v v 0}, and R = R \{v 1 v, v v+1 } = k,+1 {vk 1 v k }, where 0 n 1 and 0 m 1. Then, we can concude that R = M and R = k,+1 M k. For 0 n 1 and 0 m 1, et H = H[E E 1 M E +1 M +1 R ], the graph n H nduced by the gven unon of edge sets. Then we can see from Fgure 1 that H s a subdvson of H[E ] whch s somorphc to G. Fgure 1 shows the subgraph H and ts correspondng graph H[E ], where the edges of E +1, E, and E 1 are not drawn for carty. Aso, Fgure 1 shows that n the correspondng drawng of G for a good drawng D of H, the crossngs wthn each of v D E 1 M R and v DE +1 M +1 R need not be counted snce they are ether sef-crossngs of edges n the correspondng drawng of G or they appear on edges emanatng from the same vertex. Smary, for f, = 1, the crossngs between {v 1 v 1 } and {v +1 v +1 } need not be counted snce they both emanate from v. Therefore we have: v D H crg + v D E 1 M + v D E +1 M +1 + v D R + v D E 1 M, R + v D E +1 M +1, R + v D {v 1 v 1 }, {v +1 v +1 }. Snce f,=1 v D H = v D E + v D E 1 M + v D E +1 M +1 + v D R + v D E, E 1 M + v D E, E +1 M +1 + v D E 1 M, E +1 M +1 + v D E 1 M, R + v D E +1 M +1, R, for 0 n 1 and 0 m 1, we obtan crg v D E + v D E, E 1 M + v D + v D E 1 M, E +1 f,=1 E, E +1 M +1 + v D E, R + v D E, R M +1 v D {v 1 v 1 }, {v +1 v +1 }. 4

5 So, takng the sum of both sdes of over a yeds m crg [ v D E + v D E, E 1 M + vd E, E +1 M +1 + v D E 1 M, E +1 M +1 + vd E, R v D {v 1 v 1 }, {v +1 v +1 } ] = f,=1 [ v D E + v D E, E 1 + v D E, M + v D E, E +1 + v D E, M +1 + v D E 1, E +1 + v D E 1, M +1 + v D M, E +1 + v D M, M +1 + v D E, R f,=1 v D {v 1 v 1 }, {v +1 v +1 } ]. 3 By observng how many tmes each type of crossng s counted n the above sum, we have: v D E = m v D E, v D E, E 1 = v D E, E 1, v D E, M = p + 1 v DE, M, v D E, E +1 = v D E, E +1, v D E, M +1 = p + 1 v D E, M +1, v D E 1, M +1 v D M, E+1 v D M, M +1 v D E, R = v D E, v D E 1, M +1 = v D E 1, M +1, v D M, E +1 = v D M, E +1, v D M, M +1 R = p + 1 v D M, M +1, and = v D E, M k. k,+1 These equates and nequates, together wth the nequaty 3, resut n the foowng: m crg m v D E + [v D E, E 1 + v D E, E +1 + v D E 1, M +1 + v D M, E +1 ] + p + 1 [v D E, M + v D E, M +1 + v D M, M +1 ] + v D E 1, E +1 v D {v 1 v 1 }, {v +1 v +1 } + v D E, k,+1 M k. f,=1 Furthermore, Wenpng et a. [8] anayze dfferent types of crossngs between E 1 and E +1 and obtan the nequaty [ v D E 1, E +1 f,=1 v D {v 1 v 1 }, {v +1 v +1 } ] v D E 1, E +1. More pre- 5

6 csey, they dd not need to specfy the condton f, = 1, but the exacty same argument can be used to show the above nequaty. The detas are trva and we omt them. Therefore, for 0 n 1, we have m crg m v D E + v D E 1, E +1 + [v D E, E 1 + v D E, E +1 + v D E 1, M +1 + v D M, E +1 ] + p + 1 [v D E, M + v D E, M +1 + v D M, M +1 ] + v D E, M k, k,+1 4 so by takng the sum of both sdes of 4 over a, we get [ mn crg m v D E + v D E 1, E +1 + [v D E, E 1 + v D E, E +1 + v D E 1, M +1 + v D M, E +1 ] + p + 1 [v D E, M + v D E, M +1 + v D M, M +1 ] ] + v D E, M k = m + k,+1 v D E + v D E, M + + v D E, E v D E, E +1 v D E, M 1 + p p + 1 v D E, M + p + 1 v D E, M +1 + v D E, k,+1 v D M, M +1 M k. 5 Comparng 5 wth 1, we fnd that f p 3 and m, n 4, then mn crg m p 1 v D E + p + 1 v D H. Snce ths nequaty s aso true for the optma drawng D such that v D H = crg C n, we obtan crg C n mn p + 1 m crg p v D E mn p + 1 m crg 1n crg. p Conectures and Further Research The we-known crossng emma states that any graph G wth v vertces and e > 4v edges satsfes crg 1 e 3 64 v. Usng ths, we can fnd that crg C n {0.5p+1mn}3 64mn = Omn. However, our man 6

7 theorem predcts crg C n OcrG mn, so t s a stronger resut. It woud be nterestng to be abe to use ths theorem to obtan ower bounds for crg C n where G s the hypercube graph Q n, the reguar b- or mut- partte graph K m,m or K m,...,m, the Petersen graph, or a generazed Petersen graph, a of whch are vertex-transtve. Another nterestng exampe woud be the famy of Cartesan products of mutpe copes of somorphc cyces,.e. C n... C n, suggested by Chheon Km f possbe, ths woud be the frst genera resut concernng the crossng number of the Cartesan product of more than two graphs. We can make a conectura ower bound of crg C n for a gven vertex-transtve graph G usng a conectura vaue of crg better yet f we can cacuate crg. For exampe, et Q 3 be the cubc hypercube graph, and et P be the Petersen graph. Remember that crq 3 = 0 and crp = and that for both graphs we have p = 3. We propose the foowng conectures based on Fgure : Conecture 3.1. crq 3 = 3, and therefore crq 3 C n 6n. Conecture 3.. crp = 6, and therefore crp C n 1n. Fgure : Q 3 and P. These, f true, are stronger than the ower bounds we can fnd by usng the dsont subcyces of G: crq 3 C n 4n and crp C n 7n. A dfferent drecton of research that may be frutfu s appyng the theory of arrangements deveoped by Adamsson [1] and Adamsson and Rchter []. For an appcaton of the theory, see Geebsky and Saazar [5]. It may be possbe to use the theory of arrangement to cacuate the crossng numbers or ther ower bounds for certan sma vertex-transtve graphs G and the correspondng G, such as ones mentoned above. Fnay, consderng the ntrcate reatonshp between our functon f and the adacency matrx of G, t may be possbe to use methods from agebrac graph theory n order to expot subter symmetres n non-vertex-transtve graphs and further generaze our Man Theorem. Acknowedgements The author s gratefu to hs research mentor Chheon Km for gudng through conductng mathematca research. He aso thanks Professor Pave Etngof and Professor Jacob Fox for suggestng an nta proect concernng crossng numbers, Dr. Tanya Khovanova for genera advce on mathematca research and presentaton, and the MIT-PRIMES program for makng ths research possbe. 7

8 References [1] Adamsson J. Ph.D. Thess, Careton Unversty, 000. [] Adamsson J. and Rchter R. B., Arrangements, crcuar arrangements and the crossng number of C 7 C n, J. Combnatora Theory Seres B , [3] Deste R., Graph Theory, Sprnger 01. [4] Garey, M. R., & Johnson, D. S. Crossng number s NP-compete. SIAM Journa on Agebrac Dscrete Methods , [5] Gebsky, L. Y., & Saazar, G. The crossng number of C m C n s as conectured for n mm + 1. J. Graph Theory , [6] Kešč M. The crossng number of K,3 C 3. Dscrete mathematcs 51 00, [7] Rchter, R. B. and Sǐráň, J. The crossng number of K 3,n n a surface. J. Graph Theory , [8] Wenpng, Z., Xaohu, L., Yuansheng, Y., & Chengru, D. On the crossng numbers of K m C n and K m, P n. Dscrete Apped Mathematcs ,