An Upper Bound for the Ramsey Number of a Cycle of Length Four Versus Wheels

Surahmat,; Baskoro, Edy Tri; Uttunggadewa, Saladin; Broersma, Hajo

doi:10.1007/978-3-540-30540-8_20

Cited by 9 publications

(7 citation statements)

References 3 publications

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“…Tse [25] determined the value of R(C 4 , W n ) for 3 ≤ n ≤ 12. Surahmat et al [23] established an upper bound, which is R(C 4 , W n ) ≤ n + n/3 + 1 for n ≥ 6. Recently, Dybizbański and Dzido [10] refined the upper bound and determined some exact values of R(C 4 , W n ).…”

Section: Theorem 2 (Burr and Erdősmentioning

confidence: 99%

Three Results on Cycle-Wheel Ramsey Numbers

Zhang

Broersma

Chen

2015

Graphs and Combinatorics

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Given two graphs G 1 and G 2 , the Ramsey number R(G 1 , G 2 ) is the smallest integer N such that, for any graph G of order N , either G 1 is a subgraph of G, or G 2 is a subgraph of the complement of G. We consider the case that G 1 is a cycle and G 2 is a (generalized) wheel. We expand the knowledge on exact values of Ramsey numbers in three directions: large cycles versus wheels of odd order; large wheels versus cycles of even order; and large cycles versus generalized odd wheels.

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Section: Theorem 2 (Burr and Erdősmentioning

confidence: 99%

Three Results on Cycle-Wheel Ramsey Numbers

Zhang

Broersma

Chen

2015

Graphs and Combinatorics

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“…Surahmat et al [8] showed that R(C 4 , W m ) = 9, 10 and 9 for m = 4, 5 and 6 respectively. Independently, Kung-Kuen Tse [10] showed that R(C 4 , W m ) = 10, 9, 10, 9, 11, 12, 13, 14, 16 and 17 for m = 4, 5, 6, 7, 8, 9, 10, 11, 12 and 13, respectively.…”

Section: Every Pair Of Non-adjacent Vertices V and W Of G Then G Is mentioning

confidence: 99%

“…Independently, Kung-Kuen Tse [10] showed that R(C 4 , W m ) = 10, 9, 10, 9, 11, 12, 13, 14, 16 and 17 for m = 4, 5, 6, 7, 8, 9, 10, 11, 12 and 13, respectively. In 2005, Surahmat et al [9] obtained property that R(C 4 , W n ) ≤ n + (n − 1)/3 . Suppose that we have an admissible coloring of K m without C 4 in color 1 and without W n in color 2.…”

Section: Every Pair Of Non-adjacent Vertices V and W Of G Then G Is mentioning

confidence: 99%

On Some Ramsey Numbers for Quadrilaterals Versus Wheels

Dybizbański

Dzido

2013

Graphs and Combinatorics

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For given graphs G 1 and G 2 , the Ramsey number R(G 1 , G 2 ) is the least integer n such that every 2-coloring of the edges of K n contains a subgraph isomorphic to G 1 in the first color or a subgraph isomorphic to G 2 in the second color. Surahmat et al. proved that the Ramsey number R(C 4 , W n ) ≤ n+ (n−1)/3 . By using asymptotic methods one can obtain the following property:In this paper we show that in fact R(C 4 , W n ) ≤ n+ √ n − 2+1 for n ≥ 11. Moreover, by modification of the Erdős-Rényi graph we obtain an exact value R(C 4 , W q 2 +1 ) = q 2 +q+1 with q ≥ 4 being a prime power. In addition, we provide exact values for Ramsey numbers R(C 4 , W n ) for 14 ≤ n ≤ 17.

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“…An interesting question in this respect is: what is the best possible upper bound for R(W n , C 4 )? Surahmat et al [9] showed that R(W n , C 4 ) ≤ n + n/3 + 1 for n ≥ 6. Clearly, this upper bound is not tight in general.…”

Section: Introductionmentioning

confidence: 99%

A remark on star-C4 and wheel-C4 Ramsey numbers

Zhang¹,

Broersma²,

Chen³

2014

ejgta

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Given two graphs G 1 and G 2 , the Ramsey number R(G 1 , G 2 ) is the smallest integer N such that, for any graph G of order N , either G 1 is a subgraph of G, or G 2 is a subgraph of the complement of G. Let C n denote a cycle of order n, W n a wheel of order n + 1 and S n a star of order n. In this paper, it is shown that R(W n , C 4 ) = R(S n+1 , C 4 ) for n ≥ 6. Based on this result and Parsons' results on R(S n+1 , C 4 ), we establish the best possible general upper bound for R(W n , C 4 ) and determine some exact values for R(W n , C 4 ).

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An Upper Bound for the Ramsey Number of a Cycle of Length Four Versus Wheels

Cited by 9 publications

References 3 publications

Three Results on Cycle-Wheel Ramsey Numbers

Three Results on Cycle-Wheel Ramsey Numbers

On Some Ramsey Numbers for Quadrilaterals Versus Wheels

A remark on star-C4 and wheel-C4 Ramsey numbers

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