Monochromatic Hamiltonian t‐tight Berge‐cycles in hypergraphs |
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Authors: | Paul Dorbec Sylvain Gravier Gábor N Sárközy |
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Institution: | 1. Institut Fourier, 100 Rue des Maths, BP74, 38402 Saint Martin D'Hères, France;2. Computer Science Department, Worcester Polytechnic Institute, Worcester, Massachusetts 01609;3. Computer and Automation Research Institute, Hungarian Academy of Sciences, Budapest H‐1518, P.O. Box 63, Hungary |
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Abstract: | In any r‐uniform hypergraph for 2 ≤ t ≤ r we define an r‐uniform t‐tight Berge‐cycle of length ?, denoted by C?(r, t), as a sequence of distinct vertices v1, v2, … , v?, such that for each set (vi, vi + 1, … , vi + t ? 1) of t consecutive vertices on the cycle, there is an edge Ei of that contains these t vertices and the edges Ei are all distinct for i, 1 ≤ i ≤ ?, where ? + j ≡ j. For t = 2 we get the classical Berge‐cycle and for t = r we get the so‐called tight cycle. In this note we formulate the following conjecture. For any fixed 2 ≤ c, t ≤ r satisfying c + t ≤ r + 1 and sufficiently large n, if we color the edges of Kn(r), the complete r‐uniform hypergraph on n vertices, with c colors, then there is a monochromatic Hamiltonian t‐tight Berge‐cycle. We prove some partial results about this conjecture and we show that if true the conjecture is best possible. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 34–44, 2008 |
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Keywords: | colored complete uniform hypergraphs monochromatic hamiltonian Berge‐cycles |
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