Monochromatic Hamiltonian Berge-cycles in colored complete uniform hypergraphs
نویسندگان
چکیده
We conjecture that for any fixed r and sufficiently large n, there is a monochromatic Hamiltonian Bergecycle in every (r − 1)-coloring of the edges of K n , the complete r-uniform hypergraph on n vertices. We prove the conjecture for r = 3, n 5 and its asymptotic version for r = 4. For general r we prove weaker forms of the conjecture: there is a Hamiltonian Berge-cycle in (r−1)/2 -colorings of K n for large n; and a Berge-cycle of order (1− o(1))n in (r − log2 r )-colorings of K n . The asymptotic results are obtained with the Regularity Lemma via the existence of monochromatic connected almost perfect matchings in the multicolored shadow graph induced by the coloring of K n . © 2007 Elsevier Inc. All rights reserved.
منابع مشابه
Monochromatic Path and Cycle Partitions in Hypergraphs
Here we address the problem to partition edge colored hypergraphs by monochromatic paths and cycles generalizing a well-known similar problem for graphs. We show that r-colored r-uniform complete hypergraphs can be partitioned into monochromatic Berge-paths of distinct colors. Also, apart from 2k − 5 vertices, 2-colored k-uniform hypergraphs can be partitioned into two monochromatic loose paths...
متن کاملMonochromatic Hamiltonian 3-tight Berge cycles in 2-colored 4-uniform hypergraphs
Here improving on our earlier results we prove that there exists an n0 such that for n ≥ n0, in every 2-coloring of the edges of K n there is a monochromatic Hamiltonian 3-tight Berge cycle. This proves the c = 2, t = 3, r = 4 special case of a conjecture from [5].
متن کاملLong Monochromatic Berge Cycles in Colored 4-Uniform Hypergraphs
Here we prove that for n ≥ 140, in every 3-coloring of the edges of K (4) n there is a monochromatic Berge cycle of length at least n− 10. This result sharpens an asymptotic result obtained earlier. Another result is that for n ≥ 15, in every 2-coloring of the edges of K n there is a 3-tight Berge cycle of length at least n− 10.
متن کاملThe 3-color Ramsey number of a 3-uniform Berge-cycle
The asymptotics of 2-color Ramsey numbers of loose and tight cycles in 3-uniform hypergraphs have been recently determined ([16], [17]). We address here the same problem for Berge-cycles and for 3 colors. Our main result is that the 3-color Ramsey number of a 3-uniform Berge cycle of length n is asymptotic to 5n 4 . The result is proved with the Regularity Lemma via the existence of a monochrom...
متن کاملThe 3-Colour Ramsey Number of a 3-Uniform Berge Cycle
The asymptotics of 2-colour Ramsey numbers of loose and tight cycles in 3-uniform hypergraphs were recently determined [16, 17]. We address the same problem for Berge cycles and for 3 colours. Our main result is that the 3-colour Ramsey number of a 3-uniform Berge cycle of length n is asymptotic to 5n 4 . The result is proved with the Regularity Lemma via the existence of a monochromatic connec...
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:
- J. Comb. Theory, Ser. B
دوره 98 شماره
صفحات -
تاریخ انتشار 2008