Document Type

Article

Publication Date

8-2-2007

Abstract

In anyr-uniform hypergraph H for 2 · t · r we de¯ne 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

con- secutive 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 any

¯xed 2

· c; t · r satisfying c+t · r+1 and su±ciently large n

, if we color

the edges of K(r)

n, 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.

DOI

WPI-CS-TR-07-14

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