#### Document Type

Article

#### Publication Date

8-2-2007

#### Abstract

In any* r*-uniform hypergraph

*for 2*

*H**· t ·*we de¯ne an

*r*

*r*-uniform t -tight Berge-cycle of length * `*, denoted by

*(*

*C**)`*

*r;t**;*

as a sequence of distinct vertices

* v*1

*2*

*; v**; : : : ; v*, such that for each set (

*`**vi; v*+1

*i**; : : : ; v*+

*i**t*1) of

*¡*

*t*con- secutive vertices on the cycle, there is an edge

*E i *of

*that contains these t*

*H*vertices and the edges *E i *are all distinct for

*1*

*i;**· 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 ·*+1 and su±ciently large

*r*

*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.

#### Suggested Citation

Dorbec, Paul
, Gravier, Sylvain
, Sárközy, Gábor N.
(2007). Monochromatic Hamiltonian t-tight Berge-cycles in hypergraphs. .

Retrieved from:
https://digitalcommons.wpi.edu/computerscience-pubs/14

#### DOI

WPI-CS-TR-07-14