THE CHVÁTAL-ERDŐS CONDITION AND 2-FACTORS WITH A SPECIFIED NUMBER OF COMPONENTS

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1 Dscussones Mathematcae Graph Theory 27 (2007) THE CHVÁTAL-ERDŐS CONDITION AND 2-FACTORS WITH A SPECIFIED NUMBER OF COMPONENTS Guantao Chen Department of Mathematcs and Statstcs Georga State Unversty, Atlanta, GA 30303, USA Ronald J. Gould Department of Mathematcs and Computer Scence Emory Unversty, Atlanta, GA 30322, USA Ken-ch Kawarabayash Natonal Insttute of Informatcs Htotsubash, Chyoda-Ku, Tokyo , Japan Katsuhro Ota Department of Mathematcs, Keo Unversty Hyosh, Kohoku-Ku, Yokohama , Japan Akra Sato Department of Computer Scence, Nhon Unversty Sakurajosu , Setagaya-Ku, Tokyo , Japan Ingo Schermeyer Insttut für Dskrete Mathematk und Algebra Technsche Unverstät Bergakademe Freberg, D Freberg, Germany Partally supported by NSA grant H Partally supported by Japan Socety for the Promoton of Scence, Grant-n-Ad for Scentfc Research, by Sumtomo Foundaton and by Inoue Research Award for Young Scentsts. Partally supported by Japan Socety for the Promoton of Scence, Grant-n-Ad for Scentfc Research (C), , 2004, and The Research Grant of Nhon Unversty, College of Humantes and Scences.

2 402 G. Chen, R.J. Gould, K. Kawarabayash, K. Ota,... Abstract Let G be a 2-connected graph of order n satsfyng α(g) = a κ(g), where α(g) and κ(g) are the ndependence number and the connectvty of G, respectvely, and let r(m, n) denote the Ramsey number. The well-known Chvátal-Erdős Theorem states that G has a hamltonan cycle. In ths paper, we extend ths theorem, and prove that G has a 2-factor wth a specfed number of components f n s suffcently large. More precsely, we prove that (1) f n k r(a + 4, a + 1), then G has a 2-factor wth k components, and (2) f n r(2a + 3, a + 1) + 3(k 1), then G has a 2-factor wth k components such that all components but one have order three. Keywords: Chvátal-Erdős condton, 2-factor, hamltonan cycle, Ramsey number Mathematcs Subject Classfcaton: Prmary: 05C38; Secondary: 05C40, 05C45, 05C Introducton For a graph G, we denote by α(g) and κ(g) the ndependence number and the connectvty of G, respectvely. If the nequalty α(g) κ(g) holds, we say that G satsfes the Chvátal-Erdős condton, n vew of the followng well-known theorem. Theorem A (Chvátal-Erdős Theorem [4]). Every 2-connected graph G satsfyng α(g) κ(g) has a hamltonan cycle. If every par of nonadjacent vertces x and y n a graph G of order n satsfy deg G x + deg G y n, then we say that G satsfes Ore s condton. It s also well-known that every graph of order at least three satsfyng Ore s condton has a hamltonan cycle. Theorem B (Ore s Theorem [8]). Let G be a graph of order n 3. If deg G x + deg G y n for every par of nonadjacent vertces x and y of G, then G has a hamltonan cycle. Though the Chvátal-Erdős condton and Ore s condton look qute dfferent, Bondy [1] proved that they are not ndependent. Theorem C (Bondy [1]). Every graph of order at least three satsfyng Ore s condton satsfes the Chvátal-Erdős condton.

3 The Chvátal-Erdős Condton and 2-Factors wth A hamltonan cycle s a 2-factor wth exactly one component. From ths pont of vew, snce the Chvátal-Erdős condton and Ore s condton guarantee the exstence of a 2-factor wth one component, we may suspect that both condtons guarantee the exstence of a 2-factor wth a specfed number of components. Here we remark that we have to take the order of a graph nto account. For example, a balanced complete bpartte graph of order at least four satsfes both the Chvátal-Erdős condton and Ore s condton whle t has a 2-factor wth k components only f ts order s at least 4k. Therefore, when we consder ths knd of problem, we always have to consder a graph of a suffcently large order. For Ore s condton, Brandt et al. [2] proved the followng extenson of Theorem B. Theorem D (Brandt et al. [2]). Let k be a postve nteger. Then every graph of order at least 4k satsfyng Ore s condton has a 2-factor wth k components. Snce the balanced complete bpartte graph of order 4k 2 does not have a 2-factor wth k components, the bound 4k of the order n the above theorem s almost best-possble. Later, Enomoto [6] mproved the bound to 4k 1, whch s best-possble. Whle the extenson of Ore s Theorem to a 2-factor wth a specfed number of components has been studed n detal, lttle s known about the extenson of the Chvátal-Erdős Theorem n the same drecton. Thus, we present the followng conjecture. Conjecture 1. For each postve nteger k, there exsts a postve nteger f(k) such that every 2-connected graph G of order at least f(k) satsfyng α(g) κ(g) has a 2-factor wth exactly k components. Actually, Kaneko and Yoshmoto [7] tackled ths conjecture for k = 2, and almost solved t. Theorem E (Kaneko and Yoshmoto [7]). Every 4-connected graph of order at least sx satsfyng α(g) κ(g) has a 2-factor wth two components. Recently, Egawa [5] has ponted out that the proof of Theorem E n [7] msses one case to be consdered. But even f ths possble flaw s fxed, ther proof technque does not work for graphs of connectvty two or three. Therefore, they posed the followng conjecture.

4 404 G. Chen, R.J. Gould, K. Kawarabayash, K. Ota,... Conjecture F. Every 2-connected graph G of suffcently large order satsfyng α(g) κ(g) has a 2-factor wth two components. The purpose of ths paper s to gve a partal soluton to Conjecture 1. Let r(m,n) denote the Ramsey number. Theorem 2. Let k be a postve nteger and let G be a 2-connected graph wth α(g) = a κ(g). (1) If G k r(a + 4,a + 1), then G has a 2-factor wth k components. (2) If G r(2a + 3,a + 1) + 3(k 1), then G has a 2-factor wth k components such that k 1 components have order exactly three. We remark that the above theorem s not a complete soluton of Conjecture 1 snce n the conjecture the lower bound of the order n the assumpton only depends on k, whle n both (1) and (2) of Theorem 2 the lower bound depends not only on k but also on the ndependence number. In ths paper, we actually prove the followng theorem. Theorem 3. Let G be a graph of ndependence number a, and let C be a cycle n G. (1) If C k r(a + 4,a + 1), then there exst k dsjont cycles C 1,...,C k wth V (C) = k =1 V (C ). (2) If C r(2a + 3,a + 1) + 3(k 1), then there exst k dsjont cycles C 1,...,C k such that V (C) = k =1 V (C ) and C = 3 for 1 k 1. We obtan Theorem 2 by applyng Theorem 3 to a hamltonan cycle, whose exstence s guaranteed by the Chvátal-Erdős Theorem. In the next secton, we gve notaton and several defntons whch we use n the proofs. In Secton 3, we prove Theorem Termnology and Defntons For graph-theoretc termnology not explaned n ths paper, we refer the reader to [3]. We defne a walk to be a sequence of vertces for whch consecutve vertces are adjacent. We express a walk as a sequence of vertces. Let C = x 0 x 1...x l 1 x 0 be a cycle wth an mpled orentaton. We defne x + = x +1, x = x 1 and x +n = x +n (subscrpts counted modulo l).

5 The Chvátal-Erdős Condton and 2-Factors wth The subpath x x +1...x j 1 x j n C s denoted by x C xj. The same path traversed n the reverse order s denoted by x j C x. We also adopt the same notaton for a path. When there s no possblty of confuson, we sometmes consder a path and a cycle as graphs. Thus, for example, f x s a vertex n a path P, we wrte x V (P). Let H be a subgraph of a graph G, and let T be a cycle or a path n G. Then for u, v V (T), a subpath u T v s sad to be an H-cluster of T f V (u T v) V (H) and the vertces v and v +, f they exst, are not n H. In other words, an H-cluster s a maxmal subpath of T contaned n H. In ths secton, we prove Theorem Proof Proof of Theorem 3. For (1), we proceed by nducton on k. If k = 1, the concluson s trval. Suppose k 2. Let D be a cycle n G wth V (D) = V (C), and let P be a subpath of D of order r(a + 4,a + 1). Let u and v be the frst and the last vertces of P, respectvely. Also, let H be the subgraph of G nduced by V (P). Then ether there exsts a clque of order a + 4 n H, or there exsts an ndependent set of order a + 1 n H. However, snce α(h) α(g) = a < a + 1, the latter case does not occur, and hence there exsts a clque K of order a + 4 n H. Let I = {I 1,...,I l } be the set of K-clusters of P, and let x and y be the frst and the last vertces of I, respectvely (1 l). Now we choose (D,P,K) so that l, the number of K-clusters, s as small as possble. Suppose I has two K-clusters of order at least two. We may assume I 1 2 and I 2 2, and I 1 precedes I 2 along P. Let C 1 = y 1P x2 y 1 and C = x + 2 Dy 1 x + 2. Then C 1 P = r(a + 4,a + 1) and C (k 1) r(a + 4,a + 1). By the nducton hypothess, C can be decomposed nto k 1 dsjont cycles C 2,...,C k. Then C 1, C 2,...,C k gve a requred decomposton. Therefore, we may assume that P has at most one K-cluster of order two or more. We may assume that I 1 s a largest K-cluster. Then I 2,...,I l all consst of sngle vertces. Suppose I 1 5. Let C 1 = x + 1 x++ 1 x +3 1 x+ 1 and C = x 1 x +4 1 Dx1. Then C 1 = 3 and C k r(a + 4,a + 1) 3 (k 1) r(a + 4,a + 1). By the nducton hypothess, C can be decomposed nto k 1 dsjont cycles C 2,...,C k. Then C 1, C 2,...,C k gve a requred decomposton of C.

6 406 G. Chen, R.J. Gould, K. Kawarabayash, K. Ota,... Therefore, we may assume I 1 4. It follows that K = k =1 I l + 3. Snce K = a + 4, we have l a + 1. Let A = {y 1 +,...,y+ l }. Then A a + 1 > α(g) and hence A s not an ndependent set. Let y + y+ j E(G), 1 < j l. We may assume that I precedes I j along P. Let C = y j + Dy y jdy + y j + and P = u P y y j P y + y j + P v. Then V (C ) = V (D) = V (C), P s a subpath of C of order r(a + 4,a + 1) and the set of K-clusters of P s I {I,I j } {x P y y j P x }. Ths contradcts the mnmalty of the number of K-clusters, and the concluson for (1) follows. For (2), we also proceed by nducton on k. The concluson trvally holds f k = 1. Suppose k 2. Let H be the subgraph nduced by V (C). Snce H = C r(2a + 3,a + 1) + 3(k 1) > r(2a + 3,a + 1), ether there exsts a clque of order 2a + 3 n H, or there exsts an ndependent set of order a+1 n H. However, snce α(h) α(g) < a + 1, the latter case does not occur. Therefore, H has a clque K of order 2a + 3. Note that H has a hamltonan cycle C. Take a hamltonan cycle D of H so that (a) the number of K-clusters of D s as small as possble, and (b) the order of a largest K-cluster of D s as large as possble, subject to (a). Let I = {I 1,I 2,...,I l } be the set of K-clusters of D, and let x and y be the frst and the last vertces of I, respectvely (1 l). Suppose some K-cluster I has fve or more vertces. Let C 1 = x + x++ x +3 x + and C = x x +4 Dx. Then C 1 = 3 and C r(2a + 3, a + 1) + 3(k 2). By the nducton hypothess, C can be decomposed nto k 1 cycles C 2,...,C k such that C 2 = C 3 = = C k 1 = 3. Then C 1, C 2,...,C k form a requred cycle decomposton. Therefore, we may assume that I 4 for each, 1 k. We may assume that I 1 s a largest K-cluster. Assume I 3 for some, 2 l. Let D = y 1Dx x ++ Dx1 x + x+ 1 Dy1. Then D s a hamltonan cycle of H. Furthermore, D has the same number of K-clusters as D and x 1 x + x+ 1 Dy1 s a K-cluster, whch s larger than I 1. Ths contradcts the choce (b) of D. Therefore, I 2 for each, 2 l. It follows that 2a + 3 = K = l =1 I 2l + 2, whch mples l a + 1. Then by the same argument as n the last part of the proof of (1), we have a hamltonan cycle C of H such that the number of K-clusters of C s l 1. Ths contradcts the choce (a) of D. Therefore, the concluson follows.

7 The Chvátal-Erdős Condton and 2-Factors wth References [1] A. Bondy, A remark on two suffcent condtons for Hamlton cycles, Dscrete Math. 22 (1978) [2] S. Brandt, G. Chen, R. Faudree, R. Gould and L. Lesnak, Degree condtons for 2-factors, J. Graph Theory 24 (1997) [3] G. Chartrand and L. Lesnak, Graphs & Dgraphs (3rd ed.) (Wadsworth & Brooks/Cole, Monterey, CA, 1996). [4] V. Chvátal and P. Erdős, A note on hamltonan crcuts, Dscrete Math. 2 (1972) [5] Y. Egawa, personal communcaton. [6] H. Enomoto, On the exstence of dsjont cycles n a graph, Combnatorca 18 (1998) [7] A. Kaneko and K. Yoshmoto, A 2-factor wth two components of a graph satsfyng the Chvátal-Erdős condton, J. Graph Theory 43 (2003) [8] O. Ore, Note on Hamlton crcuts, Amer. Math. Monthly 67 (1960) 55. Receved 1 March 2006 Revsed 23 July 2007 Accepted 23 July 2007