A multipartite Ramsey number for odd cycles
نویسنده
چکیده
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. keywords: cycles, Ramsey number, Regularity Lemma, stability, multipartite
منابع مشابه
Multipartite Ramsey numbers for odd cycles
In this paper we study multipartite Ramsey numbers for odd cycles. We formulate the following conjecture: 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 Cn, ∗2000 Mathematics Subject Classification: 05C55, 05C38. †Research supported in part by the Nat...
متن کاملOn size multipartite Ramsey numbers for stars versus paths and cycles
Let Kl×t be a complete, balanced, multipartite graph consisting of l partite sets and t vertices in each partite set. For given two graphs G1 and G2, and integer j ≥ 2, the size multipartite Ramsey number mj(G1, G2) is the smallest integer t such that every factorization of the graph Kj×t := F1 ⊕ F2 satisfies the following condition: either F1 contains G1 or F2 contains G2. In 2007, Syafrizal e...
متن کاملOn the Ramsey Multiplicity of the Odd Cycles
The Ramsey multiplicity R(G) of a graph G is the minimum number of monochromatic copies of G in any two-colouring of the edges of Kr(G), where r(G) denotes the Ramsey number of G. Here we prove that odd cycles have super-exponentially large Ramsey multiplicity: If Cn is an odd cycle of length n, then logR(Cn) = Θ(n logn).
متن کاملGeneralized Ramsey Theory for Multiple Colors
In this paper, we study the generalized Ramsey number r(G, , . . ., Gk) where the graphs GI , . . ., Gk consist of complete graphs, complete bipartite graphs, paths, and cycles. Our main theorem gives the Ramsey number for the case where G 2 , . . ., G,, are fixed and G, ~_C, or P,, with n sufficiently large . If among G2 , . . ., G k there are both complete graphs and odd cycles, the main theo...
متن کاملSize multipartite Ramsey numbers for stripes versus small cycles
For simple graphs G1 and G2, the size Ramsey multipartite number mj(G1, G2) is defined as the smallest natural number s such that any arbitrary two coloring of the graph Kj×s using the colors red and blue, contains a red G1 or a blue G2 as subgraphs. In this paper, we obtain the exact values of the size Ramsey numbers mj(nK2, Cm) for j ≥ 2 and m ∈ {3, 4, 5, 6}.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Journal of Graph Theory
دوره 71 شماره
صفحات -
تاریخ انتشار 2012