On size multipartite Ramsey numbers for stars versus paths and cycles

Transcription

1 Electroic Joural of Graph Theory ad Applicatios 5 (1) (2017), 4 50 O size multipartite Ramsey umbers for stars versus paths ad cycles Aie Lusiai 1, Edy Tri Baskoro, Suhadi Wido Saputro Combiatorial Mathematics Research Group Faculty of Mathematics ad Natural Scieces, Istitut Tekologi Badug Jala Gaesa 10 Badug, Idoesia aielusiai@studets.itb.ac.id, ebaskoro@math.itb.ac.id, suhadi@math.itb.ac.id Abstract Let K l t be a complete, balaced, multipartite graph cosistig of l partite sets ad t vertices i each partite set. For give two graphs G 1 ad G 2, ad iteger j 2, the size multipartite Ramsey umber m j (G 1, G 2 ) is the smallest iteger t such that every factorizatio of the graph K j t := F 1 F 2 satisfies the followig coditio: either F 1 cotais G 1 or F 2 cotais G 2. I 2007, Syafrizal et al. determied the size multipartite Ramsey umbers of paths P versus stars, for = 2, oly. Furthermore, Surahmat et al. (2014) gave the size tripartite Ramsey umbers of paths P versus stars, for =, 4, 5, 6. I this paper, we ivestigate the size tripartite Ramsey umbers of paths P versus stars, with all 2. Our results complete the previous results give by Syafrizal et al. ad Surahmat et al. We also determie the size bipartite Ramsey umbers m 2 (K 1,m, C ) of stars versus cycles, for, m 2. Keywords: size multipartite Ramsey umber, star, path, cycle Mathematics Subject Classificatio : 05D10, 05C55 DOI: /ejgta Permaet Affiliatio: Politekik Negeri Badug, Idoesia. Received: 15 November 2016, Revised: 14 Jauary 2017, Accepted: 10 February

2 O size multipartite Ramsey umbers for stars versus paths ad cycles Aie Lusiai et al. 1. Itroductio Burger ad Vuure[1] studied oe of geeralizatios of the classical Ramsey umber problem. They itroduced the size multipartite Ramsey umber as follow. Let j, l,, s ad t be atural umbers with, s 2. The size multipartite Ramsey umber m j (K l, K s t ) is the smallest atural umber ζ such that a arbitrary colorig of the edges of K j ζ, usig the two colors red ad blue, ecessarily forces a red K l or a blue K s t as a subgraph. They also determied the exact values of m 1 (K 2 2, K 2 2 ) ad m j (K 2 2, K 1 ), for j 1. I [10], Syafrizal et al. geeralized this cocept by removig the completeess requiremet as follows. For give two graphs G 1 ad G 2, ad iteger j 2, the size multipartite Ramsey umber m j (G 1, G 2 ) = t is the smallest iteger such that every factorizatio of graph K j t := F 1 F 2 satisfies the followig coditio: either F 1 cotais G 1 as a subgraph or F 2 cotais G 2 as a subgraph. They also determied the size multipartite Ramsey umbers of paths versus other graphs, especially cycles ad stars [10, 11, 12]. I this paper, we determie the size multipartite Ramsey umbers, m j (K 1,m, H), for j = 2,, where H is a path or a cycle o vertices, ad K 1,m is a star of order m + 1. Let G be a simple ad fiite graph. The ull graph is the graph with vertices ad zero edges. A matchig of a graph G is defied as a set of edges without a commo vertex. The maximum degree of G is deoted by (G), where (G) = max{d G (v) v V (G)}. The miimum degree of G is deoted by δ(g), where δ(g) = mi{d G (v) v V (G)}. A graph G of order is called Hamiltoia if it cotais a cycle of legth ad it called bipacyclic if it cotais cycles of all eve legths from 4 to. A coected graph G is said to be k-coected, if it has more tha k vertices ad remais coected wheever fewer tha k vertices are removed. A set U of vertices i a graph G is idepedet if o two vertices i U are adjacet. The maximum umber of vertices i a idepedet set of vertices of G is called idepedet umber of G ad is deoted by α(g). For two vertices x, y G, if x is adjacet to y, the we deote by x y. Otherwise, we deote by x y. I this paper, we also use the followig theorems to prove our results. Theorem 1.1. [4] If G is a graph of order ad the miimum degree of G, δ(g), the G is a 2 Hamiltoia. Theorem 1.2. [] Let G be a s-coected graph with o idepedet set of s + 2 vertices. The, G has a Hamiltoia path. Theorem 1.. [8] Let G be a balaced bipartite graph o 2 vertices. If the miimum degree of G, δ(g) +1, the G is bipacyclic Stars versus Paths Hattigh ad Heig gave the results for the size bipartite Ramsey umbers of stars versus paths, as follows. 44

3 O size multipartite Ramsey umbers for stars versus paths ad cycles Aie Lusiai et al. Theorem 2.1. [5] For positive itegers m, 2, + m 1, for m + 1, is eve, m, for m 1 + 1, is odd, m 1 0 mod( 1), m 2 (K 1,m, P ) = + m 1, for m 1 + 1, is odd, m 1 0 mod( 1), m 1, for m < + 1, , for m < For positive itegers m, 1, Christou et al. [2] determied the size bipartite Ramsey umbers of stars K 1,m versus P 2. The size multipartite Ramsey umbers of paths P versus stars was determied oly for = 2, by Syafrizal et al. [11] i Furthermore, Surahmat et al. [9] gave the size tripartite Ramsey umbers of paths P versus stars, for =, 4, 5, 6. Lusiai et al. [7] gave the size tripartite Ramsey umbers of paths P versus a disjoit uio of m copies of a star. I this sectio, we ivestigate the size tripartite Ramsey umbers of paths P versus stars, with all 2. Our results complete the previous results give by Syafrizal et al. ad Surahmat et al. Theorem 2.2. For positive itegers 2, m (K 1,2, P ) =. Proof. For = 2,, it is clear that m (K 1,2, P ) 1. To show that m (K 1,2, P ), for 4, let us cosider a factorizatio the graph K ( 1) = F 1 F 2. We choose F 1 as a matchig, the F 1 K 1,2. Sice V (K ( 1) ) = V (F 2 ) = ( 1) <, we obtai F 2 P. Now, we show that m (K 1,2, P ). For = 2,, we kow that i ay red-blue colorig avoidig a red K 1,2, there will be a blue P 2 or a blue P. Therefore, m (K 1,2, P ) 1, for = 2,. For 4, we cosider a factorizatio K = G 1 G 2 such that G 1 does ot cotai K 1,2, so (G 1 ) 1. The δ(g 2 ) V (G 2 ) (G 1) = 2 1 = V (G 2). By Theorem , we have that G 2 is Hamiltoia which implies G 2 P, for 4. Theorem 2.. For positive iteger 2, m (K 1,, P ) = { + 1, if 2 6,, if 7. Proof. 2, if 2, To show that m (K 1,, P ) t, let t =, if 4 6,, if 7. We cosider a factorizatio the graph K (t 1) = F 1 F 2, where F 1 does ot cotai K 1,. We cosider the followig three cases. 45

4 O size multipartite Ramsey umbers for stars versus paths ad cycles Aie Lusiai et al. Case 1. For 2. We have K (t 1) = K. We ca choose F 1 = K, which implies F 1 K 1, ad F 2 P. Case 2. For 4 6. We have K (t 1) = K 2. We ca choose F 1 = C 6 ad F 2 = 2C, see Figure 1. The, F 1 K 1, ad the logest path i F 2 is a P. Figure 1. F 1 is a C 6 ad F 2 is 2C. Case. For 7. We have K (t 1) = K ( 1). We ca choose F 1 = C ( 1), the F 1 K 1,. Sice V (K (t 1) ) = V (F 2 ) = ( 1) <, we obtai F 2 P. 2, if 2, Now, we show that m (K 1,, P ) t, let t =, if 4 9,, if 10. We cosider a factorizatio K t = G 1 G 2 such that G 1 does ot cotai K 1,, so (G 1 ) 2. We cosider the followig three cases. Case 1. For 2. We have K t = K 2. Sice (G 1 ) 2, the δ(g 2 ) V (G 2 ) t (G 1 ) = = 2, which implies that G 2 P. Case 2. For 4 9. We have K t = K. Sice (G 1 ) 2, the δ(g 2 ) V (G 2 ) t (G 1 ) = 9 2 = 4. We will use Theorem 1.2 to show that G 2 is a Hamiltoia path. So, we will show that G 2 is a 2-coected graph with o idepedet set of 4 vertices. Let A, B, C be the three partities of G 2. Let x y, where x, y be ay vertices i G 2 ad S = N(x) N(y). There are two possibilities: 1. Let x ad y be i the same partite set. Sice δ(g 2 ) 4, the S ad S 2. So, two vertices of S together with x ad y form a C Let x ad y be i the differet partite sets, say x A ad y B. (a) x y. If S =, the there exist c 1, c 2 C, c 1 c 2 such that x c 1 ad y c 2. Now, sice δ(g 2 ) 4, K = N(c 1 ) N(c 2 ), say b 2 K. The {x, y, c 1, c 2, b 2 } form a C 5. Also, If S, the x ad y will be cotaied i a C. 46

5 O size multipartite Ramsey umbers for stars versus paths ad cycles Aie Lusiai et al. (b) x y. Sice δ(g 2 ) 4, the S 1 ad S C. If S 2, the x, y ad two vertices of S will create a C 4. If S = 1, the B {y} N(x) ad N(y) C = 2. Let N(y) C = {c 1, c 2 }. Sice δ(g 2 ) 4, the N(c 1 ) (B {y}) 1. Therefore, select b 1 B {y} such that c 1 b 1. The, xc 2 yc 1 b 1 x is a C 5. Sice ay two differet vertices i G 2 belogs to a cycle, G 2 is a 2-coected graph. Now, we show that α(g 2 ) =. Sice G 2 is a factor of K, α(g 2 ). α(g 2 ), as ay idepedet set of G 2 ca have at most oe elemet from each of the three partite sets. So, we have α(g 2 ) =. The, by Theorem 1.2, G 2 is a Hamiltoia path, which implies G 2 P, for 4 9. Case. For 10. We have K t = K. Sice (G 1 ) 2, the δ(g 2 ) V (G 2 ) t (G 1 ) = 2 2 = V (G 2). Thus, by Theorem 1.1, G is Hamiltoia which implies G 2 P, for 10. Theorem 2.4. For positive itegers 4 m ad 16, m (K 1,m, P ) =. Proof. To show that m (K 1,m, P ), let us cosider a factorizatio graph K ( 1) = F 1 F 2, where F 1 does ot cotai K 1,m. We ca choose F 1 = C, the F 1 K 1,m. Sice V (K ( 1) ) = V (F 2 ) = <, we obtai F 2 P. Now, we show that m (K 1,m, P ). We cosider a factorizatio K = G 1 G 2 such that G 1 does ot cotai K 1,m, so (G 1 ) m 1. The, δ(g 2 ) V (G 2 ) (G 1) = 2 (m 1). Sice δ(g 2) 2 (m 1) ad 2(m 1), the δ(g 2) 2 1 = 2 = V (G 2). The, by Theorem 1.1, G is Hamiltoia which implies G 2 P.. Stars versus Cycles The size multipartite Ramsey umbers for paths versus cycles of three or four vertices have bee showed by Syafrizal et al. [12]. Recetly, Lusiai et al. [6] showed the size multipartite Ramsey umbers for stars versus cycles. Now, we ivestigate the size bipartite Ramsey umbers for stars versus cycles. The research is ispired by the work of Hattigh ad Heig o the size bipartite Ramsey umbers for stars versus paths. It seems that m 2 (K 1,m, C ) is related to m 2 (K 1,m, P ). However, sice a complete bipartite graph does ot cotai odd cycles, the it is clear that m 2 (K 1,m, C ) =. Now, we oly cosider m 2 (K 1,m, C ), where is eve. To show this relatio, i Theorem.1, we obtai the exact value of m 2 (K 1,m, C ) for certai values of. Theorem.1. Let m 2 ad 2m, where is eve. The, { 2m 1, for 2m 4m 4, m 2 (K 1,m, C ) =, for 4m 2. 2 { 2m 1, for 2m 4m 4, Proof. Let t =, for 4m

6 O size multipartite Ramsey umbers for stars versus paths ad cycles Aie Lusiai et al. To show that m 2 (K 1,m, C ) t, let us cosider a factorizatio the graph K 2 (t 1) = F 1 F 2, such that F 1 does ot cotai K 1,m. The, (F 1 ) (m 1). We cosider the followig two cases. Case 1. For 2m 4m 4. We have K 2 (t 1) = K 2 (2m 2). We ca choose F 1 = 2K 2 (m 1). The complemet of F 1 relative to K 2 (2m 2) is 2K 2 (m 1). So, we get F 2 = 2K 2 (m 1), which implies F 1 K 1,m ad F 2 C for 2m 4m 4. Case 2. For 4m 2. We have K 2 (t 1) = K 2 ( 2 1). If we choose F 2 = K 2 ( 2 1), the F 1 is a ull graph. So, F 1 K 1,m. Sice V (K 2 (t 1) ) = V (F 2 ) = 2( 1) <, we obtai F 2 2 C. Now, we show that m 2 (K 1,m, C ) t. We cosider a factorizatio K 2 t = G 1 G 2 such that G 1 does ot cotai K 1,m, so (G 1 ) (m 1). We cosider the followig two cases. Case 1. For 2m 4m 4. We have K 2 t = K 2 (2m 1). Sice (G 1 ) (m 1), the δ(g 2 ) V (G 2 ) t (G 1 ) = 2m 1 (m 1) = m. The, by Theorem 1.4, G 2 is bipacyclic, which implies G 2 C, for 2m 4m 4. Case 2. For 4m 2. We have K 2 t = K 2 ( 2 ). Sice (G 1 ) (m 1), the δ(g 2 ) V (G 2 ) t (G 1 ) = (m 1) 1( + 1), for 4m 2. Thus, by Theorem 1., G is bipacyclic, which implies G 2 C, for 4m 2. I the ext two theorem, we cosider m 2 (K 1,m, C ) for certai values of m ad which are ot icluded i Theorem.1. I particular, we prove that m 2 (K 1,, C 4 ) = 5 i Theorem.2 ad m 2 (K 1,4, C 4 ) = 6 i Theorem.. Theorem.2. m 2 (K 1,, C 4 ) = 5. Figure 2. F 1 is a C 8 ad F 2 does ot cotai a C 4. 48

7 O size multipartite Ramsey umbers for stars versus paths ad cycles Aie Lusiai et al. Proof. To show that m 2 (K 1,, C 4 ) 5, let us cosider a factorizatio the graph K 2 4 = F 1 F 2, where F 1 does ot cotai K 1,. If we choose F 1 = C 8, the F 2 does ot cotai a C 4, as show i Figure 2. This implies that F 1 K 1, ad F 2 C 4. Now, we show that m 2 (K 1,, C 4 ) 5. We cosider a factorizatio K 2 5 = G 1 G 2 such that G 1 does ot cotai K 1,, so (G 1 ) 2. The, δ(g 2 ) V (G 2 ) t (G 1 ) = =. Thus, by Theorem 1., G 2 is bipacyclic, which implies G 2 C 4. Theorem.. m 2 (K 1,4, C 4 ) = 6. Proof. To show that m 2 (K 1,4, C 4 ) 6, let us cosider a factorizatio the graph K 2 5 = F 1 F 2, where F 1 does ot cotai K 1,4. The, (F 1 ) ad δ(f 2 ) V (F 2 ) t (F 1 ) = 10 5 = 2. We ca choose F 2 = C 10. So, we get F 2 C 4. Now, we show that m 2 (K 1,4, C 4 ) 6. We cosider a factorizatio K 2 6 = G 1 G 2 such that G 1 does ot cotai K 1,4, so (G 1 ). The, δ(g 2 ) V (G 2 ) t (G 1 ) = 12 6 =. Let A ad B be the two partitie sets of K 2 6. Let a 1 A be adjacet to b i B, i {1, 2, } i G 2. Sice δ(g 2 ), the each b i is adjacet to at least two vertices i A {a 1 }. By the Pigeohole Priciple, there exists at least oe vertex i A {a 1 } adjacet to two vertices i {b 1, b 2, b }. So, we get C 4 i G 2. Ackowledgmet This research was supported by Research Grat: Program Riset da Iovasi KK-ITB, Istitut Tekologi Badug, Miistry of Research, Techology ad Higher Educatio, Idoesia. Refereces [1] A.P. Burger ad J.H. va Vuure, Ramsey umbers i complete balaced multipartite graphs. Part II: Size Number. Discrete Math. 28 (2004), [2] M. Christou, C.S. Iliopoulos, ad M. Miller, Bipartite Ramsey umbers ivolvig stars, stripes ad trees. Electro. J. Graph Theory 1 (2) (201), [] V. Chvátal ad P. Erdös, A ote o Hamilto circuits, Discrete Math. 2 (1972), [4] G.A. Dirac, Some theorems o abstract graphs, Proc. Lod. Math. Soc. 2 (1952), [5] J.H. Hattigh ad M.A. Heig, Star-Path bipartite Ramsey umbers, Discrete Math. 185 (1998), [6] A. Lusiai, Syafrizal Sy, E.T. Baskoro, ad C. Jayawardee, O size multipartite Ramsey umbers for stars versus cycles, Procedia Comput. Sci. 74 (2015), [7] A. Lusiai, E.T. Baskoro, ad S.W. Saputro, O size tripartite Ramsey umbers of P versus mk 1,, AIP Cof. Proc (2016); doi:10.106/ [8] E. Schmeichel ad J. Mitche, Bipartite graph with cycles of all eve legths, J. Graph Theory 6 (1982),

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