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
· 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
We prove some partial results about this conjecture and we show that if true the conjecture is best possible.
, Gravier, Sylvain
, Sárközy, Gábor N.
(2007). Monochromatic Hamiltonian t-tight Berge-cycles in hypergraphs. .
Retrieved from: http://digitalcommons.wpi.edu/computerscience-pubs/14