Monochromatic Hamiltonian t-tight Berge-cycles in hypergraphs

In any r-uniform hypergraph H for 2 ≤ t ≤ r we define an runiform t-tight Berge-cycle of length , denoted by C , 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 H 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 Contract grant sponsor: National Science Foundation; Contract grant number: DMS-0456401; Contract grant sponsor: OTKA; Contract grant number: K68322. Journal of Graph Theory © 2008 Wiley Periodicals, Inc.