A well-known special case of a conjecture attributed to Ryser (actually appeared in the thesis of Henderson (1971)) states that k-partite intersecting hypergraphs have transversals of at most k−1 vertices. An equivalent form of the conjecture in terms of coloring of complete graphs is formulated in Gyárfás (1977): 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 forming a connected graph in some color. It turned out that the analogue of the conjecture for hypergraphs can be answered: it was proved in Király (2013) that in every k-coloring of the edges of the r-uniform complete hypergraph Kr (r≥3), the vertex set of Kr can be covered by at most ⌈k∕r⌉ sets, each forming a connected hypergraph 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 every color used in the coloring. We propose the following analogue of Ryser's conjecture. In every spanning(r+t)-coloring of the edges of a completer-uniformr-partite hypergraph, the vertex set can be covered by at mostt+1 sets, each forming a connected hypergraph in some color. We show that the conjecture (if true) is best possible. 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 forming a connected hypergraph 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 (where 1≤ℓ≤r). Extending our results achieved for ℓ=1, we prove that for any r≥3,2≤ℓ≤r,k≥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+⌊[formula ommited]⌋ sets, each forming a connected hypergraph in some color.
- Monochromatic component
- Ryser's conjecture
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics