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The Ramsey number of paths with respect to wheels
Discrete Mathematics 294 (2005) 275 – 277 www.elsevier.com/locate/disc
The Ramsey number of paths with respect to wheels Edy Tri Baskoroa , Surahmatb a Department of Mathematics, Institut Teknologi Bandung, Jalan Ganesa 10 Bandung, Indonesia b Department of Mathematics Education, Islamic University of Malang, Jalan MT Haryono 193 Malang 65144,
Indonesia Received 14 December 2001; received in revised form 4 November 2002; accepted 21 October 2004 Available online 31 March 2005
Abstract For graphs G and H , the Ramsey number R(G, H ) is the smallest positive integer n such that every graph F of order n contains G or the complement of F contains H . For the path Pn and the wheel Wm , it is proved that R(Pn , Wm ) = 2n − 1 if m is even, m 4, and n (m/2)(m − 2), and R(Pn , Wm ) = 3n − 2 if m is odd, m 5, and n (m − 1/2)(m − 3). © 2005 Elsevier B.V. All rights reserved. Keywords: Ramsey number; Path; Wheel
1. Introduction For graphs G and H the Ramsey number R(G, H ) is defined as the smallest n such that every graph F of order |F | = n contains G or its complement, F , contains H . Let Pn be a path with n vertices and let Wm be a wheel of m + 1 vertices; namely, a graph consists of a cycle Cm with one additional vertex being adjacent to all vertices of Cm . We determine the following values for R(Pn , Wm ). Theorem 1.
m 2n − 1 if m is even, m 4, n (m − 2), 2 R(Pn , Wm ) = 3n − 2 if m is odd, m 5, n m − 1 (m − 3). 2
E-mail addresses: [email protected] (E.T. Baskoro), [email protected] (Surahmat). 0012-365X/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2004.10.024
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E.T. Baskoro, Surahmat / Discrete Mathematics 294 (2005) 275 – 277
It was shown in [5,1] that for m = 4, 5, 6 and 7 the values in Theorem 1 are valid even for n 3, 3, 6, and 7, respectively. Further Ramsey numbers in this context can be found in [2–4,6]. Let G(V , E) be a graph. For any set S ⊂ V (G), the induced subgraph G[S] of G by S is the maximal subgraph of G with the vertex-set S. 2. Proof of Theorem 1 To prove R(Pn , Wm ) 2n − 1 or 3n − 2 for m even or odd, respectively, we observe that no Pn is in the graph 2Kn−1 or 3Kn−1 and no Wm is in their complements, respectively. R(Pn , Wm ) 2n−1, if m is even: Let F be a graph of 2n−1 vertices and F contains no path Pn . Let L1 =(l11 , l12 , l13 , . . . , l1k−1 , l1k ) be the longest path in F. Then, zl 11 , zl 1k ∈ / E(F ) for each z ∈ V1 , where V1 = V (F )\V (L1 ). Let L2 = (l21 , l22 , l23 , . . . , l2t−1 , l2t ) be the longest path in F [V1 ]. It is clear that t k. Let V2 =V (F )\(V (L1 )∪V (L2 )). Since |V (F )|=2n−1, there exists at least one vertex w ∈ V2 which is not adjacent to all endpoints l11 , l1k , l21 , l2t . We distinguish three cases. Case 1: k < m − 2. Since n (m/2)(m − 2), we can do the following process. For each i =2, 3, 4, . . . , m/2 let Li be the longest path in F [Vi−1 ], where Vi−1 =V (F )\ i−1 j =1 V (Lj ). Then, at least one vertex w remains which is not in any Lj , j = 1, 2, . . . , m/2. Clearly, w is not adjacent to all endpoints of these Lj . Thus, vertex w and all these endpoints form a wheel Wm in F . Case 2: k m − 2 and t m − 2. For i = 1, 2, 3, . . . , (m − 4)/2 define couples Ai in path L1 are as follows: {l1i+1 , l1i+2 } for i odd, Ai = {l1k−i , l1k−i+1 } for i even. Similarly, define couples Bi in path L2 are as follows: {l2i+1 , l2i+2 } for i odd, Bi = {l2k−i , l2k−i+1 } for i even. Since t k n − 1 and |F | = 2n − 1, there exists at least one vertex w which is not in L1 and L2 . Since L1 is the longest path in F, there will exist one vertex of Ai for each i, say ai , which is not adjacent to w. Similarly, since L2 is the longest path in F \V (L1 ) there must be one vertex, say bi , in couple Bi which is not adjacent to w for each i. Then {l11 , b1 , a1 , b2 , a2 , b3 , a3 , . . . , b(m−4)/2 , a(m−4)/2 , l2t , l1k , l21 } will form a cycle Cm in F since L1 is the longest path in F. Thus, those vertices together with w will form a Wm in F . Case 3: k m − 2 and t < m − 2. Since k n − 1 (F has no Pn ), V1 has at least n vertices. Then, we use the same process as in Case 1. Now we show that R(Pn , Wm ) 3n − 2 if m is odd. Let F be a graph of 3n − 2 vertices. Assume F contains no path Pn . If L1 is the longest path in F and its endpoints are l11 and / E(F ) for any z ∈ V1 where V1 = V (F )\V (L1 ). Since |V1 | 2n − 1, l1k , then zl 11 , zl 1k ∈ n (m − 1/2)((m − 1) − 2) and by the result for the case of m even, the complement of the subgraph F [V1 ] must contain a wheel Wm−1 . Then F contains a wheel Wm formed by l11 and Wm−1 since l11 is not adjacent to any vertex in V1 .
E.T. Baskoro, Surahmat / Discrete Mathematics 294 (2005) 275 – 277
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We conclude with the conjecture that for n m, R(Pn , Wm ) = 2n − 1 if m is even, and R(Pn , Wm ) = 3n − 2 if m is odd. References [1] E.T. Baskoro, The Ramsey number of paths and small wheels, J. Indones. Math. Soc. (MIHMI) 8 (1) (2002) 13–16. [2] S.A. Burr, P. Erdös, Generalization of a Ramsey-theoretic result of Chvátal, J. Graph Theory 7 (1983) 39–51. [3] V. Chvátal, F. Harary, Generalized Ramsey theory for graphs, III. Small off-diagonal numbers, Pacific J. Math. 41 (1972) 335–345. [4] S.P. Radziszowski, Small Ramsey numbers, Electron. J. Combin. (2002) DS1.9. [5] Surahmat, E.T. Baskoro, On the Ramsey number of a path or a star versus W4 or W5 , Proceedings of the 12th Australasian Workshop on Combinatorial Algorithms, Bandung, Indonesia, 14–17 July 2001, pp. 174–179. [6] H.L. Zhou, The Ramsey number of an odd cycles with respect to a wheel, J. Math. Shuxu Zazhi (Wuhan) 15 (1995) 119–120 (in Chinese).
The Ramsey number of paths with respect to wheels Edy Tri Baskoroa , Surahmatb a Department of Mathematics, Institut Teknologi Bandung, Jalan Ganesa 10 Bandung, Indonesia b Department of Mathematics Education, Islamic University of Malang, Jalan MT Haryono 193 Malang 65144,
Indonesia Received 14 December 2001; received in revised form 4 November 2002; accepted 21 October 2004 Available online 31 March 2005
Abstract For graphs G and H , the Ramsey number R(G, H ) is the smallest positive integer n such that every graph F of order n contains G or the complement of F contains H . For the path Pn and the wheel Wm , it is proved that R(Pn , Wm ) = 2n − 1 if m is even, m 4, and n (m/2)(m − 2), and R(Pn , Wm ) = 3n − 2 if m is odd, m 5, and n (m − 1/2)(m − 3). © 2005 Elsevier B.V. All rights reserved. Keywords: Ramsey number; Path; Wheel
1. Introduction For graphs G and H the Ramsey number R(G, H ) is defined as the smallest n such that every graph F of order |F | = n contains G or its complement, F , contains H . Let Pn be a path with n vertices and let Wm be a wheel of m + 1 vertices; namely, a graph consists of a cycle Cm with one additional vertex being adjacent to all vertices of Cm . We determine the following values for R(Pn , Wm ). Theorem 1.
m 2n − 1 if m is even, m 4, n (m − 2), 2 R(Pn , Wm ) = 3n − 2 if m is odd, m 5, n m − 1 (m − 3). 2
E-mail addresses: [email protected] (E.T. Baskoro), [email protected] (Surahmat). 0012-365X/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2004.10.024
276
E.T. Baskoro, Surahmat / Discrete Mathematics 294 (2005) 275 – 277
It was shown in [5,1] that for m = 4, 5, 6 and 7 the values in Theorem 1 are valid even for n 3, 3, 6, and 7, respectively. Further Ramsey numbers in this context can be found in [2–4,6]. Let G(V , E) be a graph. For any set S ⊂ V (G), the induced subgraph G[S] of G by S is the maximal subgraph of G with the vertex-set S. 2. Proof of Theorem 1 To prove R(Pn , Wm ) 2n − 1 or 3n − 2 for m even or odd, respectively, we observe that no Pn is in the graph 2Kn−1 or 3Kn−1 and no Wm is in their complements, respectively. R(Pn , Wm ) 2n−1, if m is even: Let F be a graph of 2n−1 vertices and F contains no path Pn . Let L1 =(l11 , l12 , l13 , . . . , l1k−1 , l1k ) be the longest path in F. Then, zl 11 , zl 1k ∈ / E(F ) for each z ∈ V1 , where V1 = V (F )\V (L1 ). Let L2 = (l21 , l22 , l23 , . . . , l2t−1 , l2t ) be the longest path in F [V1 ]. It is clear that t k. Let V2 =V (F )\(V (L1 )∪V (L2 )). Since |V (F )|=2n−1, there exists at least one vertex w ∈ V2 which is not adjacent to all endpoints l11 , l1k , l21 , l2t . We distinguish three cases. Case 1: k < m − 2. Since n (m/2)(m − 2), we can do the following process. For each i =2, 3, 4, . . . , m/2 let Li be the longest path in F [Vi−1 ], where Vi−1 =V (F )\ i−1 j =1 V (Lj ). Then, at least one vertex w remains which is not in any Lj , j = 1, 2, . . . , m/2. Clearly, w is not adjacent to all endpoints of these Lj . Thus, vertex w and all these endpoints form a wheel Wm in F . Case 2: k m − 2 and t m − 2. For i = 1, 2, 3, . . . , (m − 4)/2 define couples Ai in path L1 are as follows: {l1i+1 , l1i+2 } for i odd, Ai = {l1k−i , l1k−i+1 } for i even. Similarly, define couples Bi in path L2 are as follows: {l2i+1 , l2i+2 } for i odd, Bi = {l2k−i , l2k−i+1 } for i even. Since t k n − 1 and |F | = 2n − 1, there exists at least one vertex w which is not in L1 and L2 . Since L1 is the longest path in F, there will exist one vertex of Ai for each i, say ai , which is not adjacent to w. Similarly, since L2 is the longest path in F \V (L1 ) there must be one vertex, say bi , in couple Bi which is not adjacent to w for each i. Then {l11 , b1 , a1 , b2 , a2 , b3 , a3 , . . . , b(m−4)/2 , a(m−4)/2 , l2t , l1k , l21 } will form a cycle Cm in F since L1 is the longest path in F. Thus, those vertices together with w will form a Wm in F . Case 3: k m − 2 and t < m − 2. Since k n − 1 (F has no Pn ), V1 has at least n vertices. Then, we use the same process as in Case 1. Now we show that R(Pn , Wm ) 3n − 2 if m is odd. Let F be a graph of 3n − 2 vertices. Assume F contains no path Pn . If L1 is the longest path in F and its endpoints are l11 and / E(F ) for any z ∈ V1 where V1 = V (F )\V (L1 ). Since |V1 | 2n − 1, l1k , then zl 11 , zl 1k ∈ n (m − 1/2)((m − 1) − 2) and by the result for the case of m even, the complement of the subgraph F [V1 ] must contain a wheel Wm−1 . Then F contains a wheel Wm formed by l11 and Wm−1 since l11 is not adjacent to any vertex in V1 .
E.T. Baskoro, Surahmat / Discrete Mathematics 294 (2005) 275 – 277
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We conclude with the conjecture that for n m, R(Pn , Wm ) = 2n − 1 if m is even, and R(Pn , Wm ) = 3n − 2 if m is odd. References [1] E.T. Baskoro, The Ramsey number of paths and small wheels, J. Indones. Math. Soc. (MIHMI) 8 (1) (2002) 13–16. [2] S.A. Burr, P. Erdös, Generalization of a Ramsey-theoretic result of Chvátal, J. Graph Theory 7 (1983) 39–51. [3] V. Chvátal, F. Harary, Generalized Ramsey theory for graphs, III. Small off-diagonal numbers, Pacific J. Math. 41 (1972) 335–345. [4] S.P. Radziszowski, Small Ramsey numbers, Electron. J. Combin. (2002) DS1.9. [5] Surahmat, E.T. Baskoro, On the Ramsey number of a path or a star versus W4 or W5 , Proceedings of the 12th Australasian Workshop on Combinatorial Algorithms, Bandung, Indonesia, 14–17 July 2001, pp. 174–179. [6] H.L. Zhou, The Ramsey number of an odd cycles with respect to a wheel, J. Math. Shuxu Zazhi (Wuhan) 15 (1995) 119–120 (in Chinese).