Monochromatic Hamiltonian t‐tight Berge‐cycles in hypergraphs

Abstract

Abstract In any r ‐uniform hypergraph ${\cal{H}}$ for 2 ≤ t ≤ r we define an r ‐uniform t ‐tight Berge‐cycle of length ℓ, denoted by C ℓ ( r , t ) , as a sequence of distinct vertices v 1 , v 2 , … , v ℓ , such that for each set ( v i , v i + 1 , … , v i + t − 1 ) of t consecutive vertices on the cycle, there is an edge E i of ${\cal{H}}$ that contains these t vertices and the edges E i are all distinct for i , 1 ≤ i ≤ ℓ, where ℓ + 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 K n ( 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