Dscrete Mathematcs 307 (2007) 764 768 www.elsever.com/locate/dsc Note The Erdős Pósa property for vertex- and edge-dsjont odd cycles n graphs on orentable surfaces Ken-Ich Kawarabayash a, Atsuhro Nakamoto b a Natonal Insttute of Informatcs, 2-1-2, Htotsubash, Chyoda-ku, Tokyo 101-8430, Japan b Department of Mathematcs, Faculty of Educaton and Human Scences, Yokohama Natonal Unversty, 79-2 Tokwada, Hodogaya-ku, Yokohama 240-8501, Japan Receved 14 March 2006; receved n revsed form 9 June 2006; accepted 10 July 2006 Avalable onlne 24 August 2006 Abstract We prove that for any orentable surface S and any non-negatve nteger k, there exsts an nteger f S (k) such that every graph G embeddable n S has ether k vertex-dsjont odd cycles or a vertex set A of cardnalty at most f S (k) such that G A s bpartte. Such a property s called the Erdős Pósa property for odd cycles. We also show ts edge verson. As Reed [Mangoes and blueberres, Combnatorca 19 (1999) 267 296] ponted out, the Erdős Pósa property for odd cycles do not hold for all non-orentable surfaces. 2006 Elsever B.V. All rghts reserved. Keywords: Erdős Pósa property; Odd cycles; Orentable surfaces 1. Introducton A famly F of graphs s sad to have the Erdős Pósa property f for every nteger k, there s an nteger f(k,f) such that every graph G contans ether k vertex-dsjont subgraphs each somorphc to a graph n F or a set C of at most f(k,f) vertces such that G C has no subgraph somorphc to a graph n F. The edge verson can be consdered as well. The term Erdős Pósa property arose from [2], n whch Erdős and Pósa proved that the famly of cycles has ths property. Robertson and Seymour [10] extended ths to the class of graphs havng any fxed planar graph as a mnor. Thomassen [12] proved that the famly of cycles of length 0 modulo m satsfes the Erdős Pósa property. On the other hand, for odd cycles, the stuaton s dfferent. Lovász characterzes the graphs havng no two dsjont odd cycles, usng Seymour s result on regular matrods. No such characterzaton s known for more than two odd cycles. Though the Erdős Pósa property for odd cycles does not hold n general, Reed [9] ponted out that there exsts a cubc projectve planar graph whch does not contan two edge-dsjont odd cycles, but there s nether a vertex set A nor an edge set B of a bounded cardnalty such that G A and G B are bpartte. Hence ths example shows that the Erdős Pósa property does not necessarly hold for cycles of length / 0 modulo m for some m [12]. Whle the Erdős Pósa property does not hold for vertex- and edge-dsjont odd cycles n general, these are known to hold for some classes of graphs [5,4,8,13]. Moreover, Reed proved that the Erdős Pósa property holds for vertex-dsjont odd cycles n planar graphs [9].ForasetA, let A denote the cardnalty of A. E-mal addresses: k_kent@n.ac.jp (K.-I. Kawarabayash), nakamoto@edhs.ynu.ac.jp (A. Nakamoto). 0012-365X/$ - see front matter 2006 Elsever B.V. All rghts reserved. do:10.1016/j.dsc.2006.07.008
K.-I. Kawarabayash, A. Nakamoto / Dscrete Mathematcs 307 (2007) 764 768 765 In ths paper, we prove that the Erdős Pósa property holds for vertex- and edge-dsjont odd cycles n graphs embeddable n an orentable surface, as follows. Theorem 1. For any two non-negatve ntegers g and k, there exsts an nteger f g (k) such that every graph G embeddable on the orentable surface of genus g has ether k vertex-dsjont odd cycles or a vertex set A wth A f g (k) such that G A s bpartte. Theorem 2. For any two non-negatve ntegers g and k, there exsts an nteger f g (k) such that every graph G embeddable on the orentable surface of genus g has ether k edge-dsjont odd cycles or an edge set B wth B f g (k) such that G B s bpartte. Theorems 1 and 2 for the sphere have already been proved by Reed [9] and Berge and Reed [1], respectvely. After that, Král and Voss [6] determned the exact bound for f 0 (k), namely f 0 (k) = 2k. 2. Proof of the theorems Let S g denote the orentable closed surface of genus g.anembeddng (or a map) on a closed surface F 2 means a fxed embeddng of some smple graph on F 2. A face of an embeddng G s sad to be even (resp., odd) f ts facal closed walk has even (resp., odd) length. An embeddng G s sad to be even f each face of G s even. Note that every even embeddng on the sphere s bpartte, but ths does not hold for all non-sphercal surfaces. Let F 2 be a non-sphercal surface and let l be a smple closed curve on F 2. We say that l s essental f l does not bound a 2-cell on F 2, and that l s separatng f the surface F 2 l s dsconnected. Clearly, f l s non-separatng, then l s essental. A cycle C of an embeddng G on F 2 s sad to be essental (resp., separatng)fc s essental (resp., separatng) as a smple closed curve on F 2. The face-wdth (or representatvty) of an embeddng G on a non-sphercal surface F 2, denoted fw(g), sthe mnmum number of ntersectng ponts of G and l, where l ranges over all essental closed curves on F 2. Note that the face-wdth for a plane graph cannot be defned snce the plane (or the sphere) admts no essental closed curve. For the notaton concernng graphs on surfaces, the readers should refer to [7]. Lemma 3. For any two postve ntegers k and g, there exsts an nteger N g (k) such that every embeddng G on the orentable surface S g wth fw(g) N g (k) has k dsjont homotopc non-separatng cycles. In partcular, f G s a non-bpartte even embeddng on S g, then these cycles can be taken to have odd length. Proof. We prove only the latter, snce our proof also works for the former. Let G be an even embeddng on S g. We frst observe that any two homotopc cycles of G have the same party of length. Second, there s a fnte set Ω of parwse non-homotopc essental non-separatng smple closed curves on S g such that for any even embeddng G on S g, there s a closed walk of odd length of G homotopc to some element of Ω. Thrd, for any embeddng H on S g, there s an nteger N H such that any embeddng on S g wth face-wdth at least N H has H as a surface mnor [10]. Usng the above three facts, we have only to take an embeddng H on S g wth k dsjont homotopc cycles for each element of Ω and put N g (k) = N H. Let G be a graph and let A be a vertex set or an edge set of G. We say that A s bpartzng f G A s bpartte. Proof of Theorem 1. We shall defne a functon f g (k) such that every embeddng G on S g has ether k dsjont odd cycles or a bpartzng vertex set A wth A f g (k). We use nducton on g. By the result n [9], the value f 0 (k) exsts, and hence we get the frst step of nducton when g = 0. Therefore, we assume that f g (k) exsts for any g <gand consder the case when the genus s exactly g 1. Let h = max{n g (2f g 1 (k) + 2f 0 (k) + 3), N g (k) + f g 1 (k) + f 0 (k)}, where N g (k) s the number n Lemma 3. Note that h depends only on k and g snce so are N g (k) and f g (k) wth g <g.
766 K.-I. Kawarabayash, A. Nakamoto / Dscrete Mathematcs 307 (2007) 764 768 Let f g (k) = max max g 1,g 2 >0 g 1 +g 2 =g {h 1 + f g1 (k) + f g2 (k)},h 1 + f g 1 (k). Case 1: G admts an essental smple closed curve l ntersectng G at most h 1 tmes. We may assume that l ntersects G only at vertces. Then we can take the vertex set S of G ntersected by l such that S h 1. We frst suppose that l separates S g. Then S g s separated nto two punctured orentable surfaces. Pastng a dsk to each boundary component, we obtan two non-sphercal closed orentable surfaces S g1 and S g2, where g 1,g 2 > 0 and g 1 + g 2 = g. Let G be the component of G S on S g, for = 1, 2. By the nducton hypothess, each G has ether k dsjont odd cycles, or a bpartzng vertex set of cardnalty at most f g (k). Therefore, G has ether k dsjont odd cycles or a bpartzng vertex set of cardnalty at most h 1 + f g1 (k) + f g2 (k) f g (k). Secondly, we suppose that l s non-separatng. Smlarly to the former case, G S s an embeddng on S g 1. By the nducton hypothess, G S has ether k dsjont odd cycles or a bpartzng vertex set of cardnalty at most f g 1 (k). Hence G has ether k dsjont odd cycles or a bpartzng vertex set of cardnalty at most h 1 + f g 1 (k) f g (k). Case 2: The face-wdth of G s at least h. Put l = f g 1 (k) + f 0 (k) + 1. By the defnton of h and Lemma 3, G has 2l + 1 dsjont homotopc essental nonseparatng cycles C 1,...,C 2l+1 n ths order. Let H = G V(C l+1 ), whch s embedded n S g 1. By the nducton hypothess, H has ether k dsjont odd cycles or a bpartzng vertex set A wth A f g 1 (k). In the former case, the k dsjont odd cycles n H are requred ones n G, and hence we consder the latter. Next, let us consder the annular subgraph Q of G bounded by C 1 and C 2l+1.IfQ contans k dsjont odd cycles, then we are done. Hence we may assume that Q has a bpartzng vertex set A wth A f 0 (k), snce Q s planar. Let G = G A A. We clam that G has no odd face. Suppose t has. Snce there are no odd faces n Q and H, the odd face must contan both a vertex of C l+1 and a vertex outsde Q. But ths s mpossble snce we deleted at most f 0 (k) + f g 1 (k)<l vertces n the annulus Q bounded by C 1 and C 2l+1 and any curve from a vertex of C l+1 to a vertex outsde the annulus ntersects at least l vertces n G. Hence there are no odd faces n G. If G s bpartte, then A A s a requred bpartzng vertex set of G, snce A A A + A f g 1 (k) + f 0 (k) f g (k). Hence we suppose that G s non-bpartte. Snce G s obtaned from G by removng at most f g 1 (k) + f 0 (k) vertces, the face-wdth of G s stll at least h (f g 1 (k) + f 0 (k)) N g (k). By Lemma 3, G has k dsjont odd cycles. Proof of Theorem 2. Observe that the above proof works for the edge-dsjont case f the face-wdth s large enough, snce we can apply Lemma 3 and fnd many dsjont homotopc essental cycles. So we consder the case when the face-wdth s small. The dual-wdth of G on S g, denoted dw(g), s the mnmum number of ntersectng ponts of G and l, where l ranges over all essental closed curves ntersectng G only at nner ponts of edges. (The dual-wdth of G s the length of a shortest essental cycle of the surface dual of G.) If G has small face-wdth and small dual-wdth, then we can apply nducton wth respect to g by removng a few edges, smlarly to the proof of Theorem 1. However, G can have small face-wdth but an arbtrarly large dual-wdth. We shall handle only ths case. Let v be a vertex of G and let e 1,...,e m be the edges of G ncdent to v n ths cyclc order (then deg G (v) = m). For any edge e, startng at the vertex v, put m/2 vertces v 1,...,v m/2 n ths order. (Each edge of the resultng embeddng on the path between v and v m/2 s called an auxlary edge.) Next, jon v j and v j +1 by a path of length 2 for each and j (where the ndces and + 1 are taken modulo m). We call ths operaton a patch extenson wth respect to v. (See Fg. 1, for example.) Clearly, the patch (.e., the plane graph wth outer cycle through v m/2 1,...,vm m/2 )s bpartte. Each face n a patch s called an auxlary face of the resultng embeddng. The m/2 cycles of length 2m surroundng v are called the nested cycles for v. Let G be the embeddng on S g obtaned from G by the patch extensons wth respect to all vertces of G. Note that each edge of G not on any nested cycle corresponds to some edge of G. A non-auxlary edge and face of G are sad to be ntrnsc. Clearly, the ntrnsc edges of G and the edges of G have the one-to-one correspondence, and hence so do
K.-I. Kawarabayash, A. Nakamoto / Dscrete Mathematcs 307 (2007) 764 768 767 Fg. 1. A patch extenson wth respect to v. the ntrnsc faces of G and the faces of G. Moreover, every auxlary face s even, and G and G have the same number of odd faces. Observe that the face-wdth of G s greater than or equal to the dual-wdth of G. Snce the dual-wdth of G s assumed to be large enough, so s the face-wdth of G. Therefore, G satsfes the theorem, as descrbed n the begnnng of the proof. That s, G has ether k edge-dsjont odd cycles C 1,..., C k or a bpartzng edge set S of bounded cardnalty. In order to complete the proof, we shall prove that G has k edge-dsjont odd cycles correspondng to C 1,..., C k,ora bpartzng edge set S wth S S. We frst consder the former case. It s an mportant observaton that every odd cycle of G must use an odd number of ntrnsc edges of G, snce the graph n each patch s bpartte and snce v m/2 and v m/2 +1 n the same patch belong to the same partte set of ts bpartton, for any. Hence the cycle, say C,nG correspondng to C has odd length. Snce C 1,..., C k are edge-dsjont n G,soareC 1,...,C k n G. Now we consder the latter. Let S be the set of edges of G correspondng to the edges of S whch are not on nested cycles. Then we have S S. We clam that G S s bpartte. If not, then G S has an odd cycle, say C=u 0,e 1,u 1,e 2,...,u 2l,e 2l+1,u 2l+1 (=u 0 ), where u V (G) and e E(G) for each. For each, let L be the path n G correspondng to e. By the defnton of S, no edge s deleted from L n G S. Hence, G S has the cycle C = 2l+1 =1 L correspondng to C. Snce each L conssts of one ntrnsc edge and deg G (u 1 )/2 + deg G (u )/2 auxlary edges n G, 2l+1 C = =1 2l+1 L =2l + 1 + =1 degg (v ) 2 1 (mod 2). 2 Ths contradcts that G S s bpartte. As s mentoned n Secton 1, t has been proved by Král and Voss [6] that f 0 (k) = 2k. Let us estmate f 1 (k) for the torus S 1, usng the result of de Graaf and Schrjver [3],.e., every torodal embeddng G wth fw(g) 2 3 r has a torodal grd C r C r as a surface mnor, and hence G has r dsjont homotopc essental cycles. Proposton 4. f 1 (k) 14k + 4. Proof. Let G be any embeddng on the torus S 1. We may suppose that fw(g) dw(g), as n the proof of Theorem 2. Suppose that dw(g) 12k + 5. (For otherwse, removng at most 12k + 4 edges, we obtan a plane graph, whch has a bpartzng edge set of cardnalty 2k, by Král and Voss s result [6]. Therefore, G has a bpartzng edge set of cardnalty 12k + 4 + 2k = 14k + 4.) Snce fw(g) 12k + 5, G has 8k + 3 dsjont homotopc essental cycles C 1,...,C 8k+3 by the above mentoned de Graaf and Schrjver s result. Then, consderng two annular subgraphs H = G V(C 4k+2 ) and Q bounded by C 1 and C 8k+3 and removng at most 2k + 2k = 4k edges from G, we get an even embeddng G from G. (If G has an odd face f, then f must have a vertex of C 4k+2 and a vertex not contaned n Q. However, ths s mpossble, snce we deleted at
768 K.-I. Kawarabayash, A. Nakamoto / Dscrete Mathematcs 307 (2007) 764 768 most 4k edges. The detals should be referred to the proof of Theorem 2.) If G s bpartte, then the edge removed s a requred bpartzng edge set of cardnalty at most 4k<14k + 5. Hence we suppose that G s non-bpartte. Snce fw(g ) 12k + 5 4k = 8k + 5, G has C 5k+3 C 5k+3 as a surface mnor, by the above result. Let A 1,...,A 5k+3 be 5k + 3 dsjont homotopc essental cycles n G correspondng to those n C 5k+3 C 5k+3, Note that A 1,...,A 5k+3 have the same party of length, snce they are homotopc and snce G s an even embeddng. Moreover, G has another set of dsjont homotopc essental cycles orthogonal to A s, snce G has C 5k+3 C 5k+3 as a surface mnor. Let B 1,...,B 5k+3 be such cycles, whch have the same party of length. Observe that A 1 or B 1 have odd length, snce A 1 and B 1 cut the torus nto a dsk and snce G s non-bpartte. Hence G has 5k + 3 ( k) odd cycles. In order to get a lnear bound for f g (k) wth g 2, t suffces to prove that the face-wdth bounded by a lnear functon of k guarantees the exstence of k dsjont homotopc cycles wth a specfed homotopy type on S g,asntheabove proof. However, t does not seem to be easy. Acknowledgements We are grateful to the anonymous two referees to gve us helpful comments pontng out our small mstakes and suggestons mprovng our expressons. References [1] C. Berge, B. Reed, Optmal packng of edge-dsjont odd cycles, Dscrete Math. 211 (2000) 197 202. [2] P. Erdös, L. Posá, On the maxmal number of dsjont crcuts of a graph, Publ. Math. Debrecen 9 (1962) 3 12. [3] M. de Graaf, A. Schrjver, Grd mnors of the graphs on the torus, J. Combn. Theory Ser. B 61 (1994) 57 62. [4] K. Kawarabayash, B. Reed, Hghly party lnked graphs, Combnatorca, to appear. [5] K. Kawarabayash, P. Wollan, Non-zero dsjont cycles n hghly connected group labelled graphs, J. Combn. Theory Ser. B. 96 (2006) 296 301. [6] D. Král, H.J. Voss, Edge-dsjont odd cycles n planar graphs, J. Combn. Theory Ser. B 90 (2004) 107 120. [7] B. Mohar, C. Thomassen, Graphs on Surfaces, Johns Hopkns Unversty Press, Baltmore, MD, 2001. [8] D. Rautenbach, B. Reed, The Erdős Pósa property for odd cycles n hghly connected graphs, Combnatorca 21 (2001) 267 278. [9] B. Reed, Mangoes and blueberres, Combnatorca 19 (1999) 267 296. [10] N. Robertson, P. Seymour, Graph mnors. V. Excludng a planar graph, J. Combn. Theory Ser. B 41 (1986) 92 114. [12] C. Thomassen, On the presence of dsjont subgraphs of a specfed type, J. Graph Theory 12 (1988) 101 111. [13] C. Thomassen, The Erdős Pósa property for odd cycles n graphs of large connectvty, Combnatorca 21 (2001) 321 333. Further readng [11] N. Robertson, P. Seymour, Graph mnors. VII. Dsjont paths on a surface, J. Combn. Theory, Ser. B 45 (1988) 212 254.