Monochromatic Hamiltonian t ‐tight Berge‐cycles in hypergraphs

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