A Note on Odd Cycle-Complete Graph Ramsey Numbers

Discrete Mathematics, 1973

For m, n 13, the cycle Ramsey number c (m, n) is the least integer p such that for any graph G of order p, either G contains an m-cycle, or its cmmplement c contains an n-cycle. Lower bounds are established for c(m, n) and these bounds lead ehe authors to conjecture a formula for the precise values bt' all c (m, n). Aiso, for m odd and m s_ n, the lower bound for c(m, n) is coupled with a result of Gondy to determine that c(rn, nz) = 2rr-1 for all odd 1112 5.

Discussiones Mathematicae Graph Theory, 2005

The cycle-complete graph Ramsey number r(C m , K n) is the smallest integer N such that every graph G of order N contains a cycle C m on m vertices or has independence number α(G) ≥ n. It has been conjectured by Erdős, Faudree, Rousseau and Schelp that r(C m , K n) = (m − 1)(n − 1) + 1 for all m ≥ n ≥ 3 (except r(C 3 , K 3) = 6). This conjecture holds for 3 ≤ n ≤ 6. In this paper we will present a proof for r(C 5 , K 7) = 25.

ISRN Algebra, 2011

It has been conjectured by Erdős, Faudree, Rousseau, and Schelp that for all satisfying that (except . In this paper, we prove this for the case and .

1979

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz COMMENTATIONES MATHEMATICAE UNIVERSITATIS CAROUNAE 20,3 (1979) ON RAMSEY GRAPHS WITHOUT CYCLES OF SHORT ODD LENGTHS J. NESETfclL, V. RODL Abstract; The Ramsey problem for classes of graphs without short odd cycles is solved. The proof uses category theoretical means.

Graphs and Combinatorics, 2008

For given graphs G and H, the Ramsey number R(G, H) is the smallest positive integer N such that for every graph F of order N the following holds: either F contains G as a subgraph or the complement of F contains H as a subgraph. In this paper, we determine the Ramsey number R(C n , W m ) = 3n − 2 for odd m ≥ 5 and \(n > \frac{5m-9}{2}\) .

European Journal of Combinatorics, 2010

For two given 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 contains G 1 or the complement of G contains G 2 . Let C n denote a cycle of order n and W m a wheel of order m + 1. Surahmat, Baskoro and Tomescu conjectured that R(C n , W m ) = 3n − 2 for m odd, n ≥ m ≥ 3 and (n, m) = (3, 3). In this paper, we confirm the conjecture for n ≥ 20.

Journal of Graph Theory, 2012

In this paper we study multipartite Ramsey numbers for odd cycles. Our main result is the proof of a conjecture of Gyárfás, Sárközy and Schelp [12]. Precisely, let n ≥ 5 be an arbitrary positive odd integer; then in any two-coloring of the edges of the complete 5-partite graph K (n−1)/2,(n−1)/2,(n−1)/2,(n−1)/2,1 there is a monochromatic cycle of length n.

Transactions of the American Mathematical Society, 1972

Let c ( m , n ) c(m,n) be the least integer p p such that, for any graph G G of order p p , either G G has an m m -cycle or its complement G ¯ \bar G has an n n -cycle. Values of c ( m , n ) c(m,n) are established for m , n ⩽ 6 m,n \leqslant 6 and general formulas are proved for c ( 3 , n ) , c ( 4 , n ) c(3,n),c(4,n) , and c ( 5 , n ) c(5,n) .

Electronic Notes in Discrete Mathematics, 2005

For graphs L 1 , . . . , L k , the Ramsey number R(L 1 , . . . , L k ) is the minimum integer N satisfying that for any coloring of the edges of the complete graph K N on N vertices by k colors there exists a color i for which the corresponding color class contains L i as a subgraph.

Journal of Combinatorial Theory, Series B, 2009

Denote by R(L, L, L) the minimum integer N such that any 3-coloring of the edges of the complete graph on N vertices contains a monochromatic copy of a graph L. Bondy and Erdős conjectured that when L is the cycle C n on n vertices, R(C n , C n , C n ) = 4n − 3 for every odd n > 3. Luczak proved that if n is odd, then R(C n , C n , C n ) = 4n + o(n), as n → ∞, and Kohayakawa, Simonovits and Skokan confirmed the Bondy-Erdős conjecture for all sufficiently large values of n.