2 research outputs found
Vertex covers by monochromatic pieces - A survey of results and problems
This survey is devoted to problems and results concerning covering the
vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles
and other objects. It is an expanded version of the talk with the same title at
the Seventh Cracow Conference on Graph Theory, held in Rytro in September
14-19, 2014.Comment: Discrete Mathematics, 201
Covering complete partite hypergraphs by monochromatic components
A well-known special case of a conjecture attributed to Ryser states that
k-partite intersecting hypergraphs have transversals of at most k-1 vertices.
An equivalent form was formulated by Gy\'arf\'as: if the edges of a complete
graph K are colored with k colors then the vertex set of K can be covered by at
most k-1 sets, each connected in some color. It turned out that the analogue of
the conjecture for hypergraphs can be answered: Z. Kir\'aly proved that in
every k-coloring of the edges of the r-uniform complete hypergraph K^r (r >=
3), the vertex set of K^r can be covered by at most sets,
each connected in some color.
Here we investigate the analogue problem for complete r-uniform r-partite
hypergraphs. An edge coloring of a hypergraph is called spanning if every
vertex is incident to edges of any color used in the coloring. We propose the
following analogue of Ryser conjecture.
In every spanning (r+t)-coloring of the edges of a complete r-uniform
r-partite hypergraph, the vertex set can be covered by at most t+1 sets, each
connected in some color.
Our main result is that the conjecture is true for 1 <= t <= r-1. We also
prove a slightly weaker result for t >= r, namely that t+2 sets, each connected
in some color, are enough to cover the vertex set.
To build a bridge between complete r-uniform and complete r-uniform r-partite
hypergraphs, we introduce a new notion. A hypergraph is complete r-uniform
(r,l)-partite if it has all r-sets that intersect each partite class in at most
l vertices.
Extending our results achieved for l=1, we prove that for any r >= 3, 2 <= l
= 1+r-l, in every spanning k-coloring of the edges of a complete
r-uniform (r,l)-partite hypergraph, the vertex set can be covered by at most
1+\lfloor \frac{k-r+\ell-1}{\ell}\rfloor sets, each connected in some color.Comment: 14 page