It is well known that every bipartite graph with vertex classes of size nwhose minimum degree is at least n/2 contains a perfect matching. We prove an analogue of this result for uniform hypergraphs. We also provide an analogue of Dirac’s theorem on Hamilton cycles for 3-uniform hypergraphs: We say that a 3-uniform hypergraph has a Hamilton cycle if there is a cyclic ordering of its vertices su...

We prove that coloring a 3-uniform 2-colorable hypergraph with c colors is NP-hard for any constant c. The best known algorithm [20] colors such a graph using O(n1/5) colors. Our result immediately implies that for any constants k ≥ 3 and c2 > c1 > 1, coloring a k-uniform c1-colorable hypergraph with c2 colors is NP-hard; the case k = 2, however, remains wide open. This is the first hardness re...

A uniform hypergraph H is called k-Ramsey for a hypergraph F , if no matter how one colors the edges of H with k colors, there is always a monochromatic copy of F . We say that H is minimal k-Ramsey for F , if H is k-Ramsey for F but every proper subhypergraph of H is not. Burr, Erdős and Lovasz [S. A. Burr, P. Erdős, and L. Lovász, On graphs of Ramsey type, Ars Combinatoria 1 (1976), no. 1, 16...

A hypergraph H is a sum hypergraph iff there are a finite S ⊆ IN and d, d ∈ IN with 1 < d ≤ d such that H is isomorphic to the hypergraph Hd,d(S) = (V, E) where V = S and E = {e ⊆ S : d ≤ |e| ≤ d ∧ v∈e v ∈ S}. For an arbitrary hypergraph H the sum number σ = σ(H) is defined to be the minimum number of isolated vertices y1, . . . , yσ 6∈ V such that H ∪ {y1, . . . , yσ} is a sum hypergraph. Gene...

Using a generalisation of Hamiltonian cycles to uniform hypergraphs due to Katona and Kierstead, we define a new notion of a Hamiltonian decomposition of a uniform hypergraph. We then consider the problem of constructing such decompositions for complete uniform hypergraphs, and describe its relationship with other topics, such as design theory.

The competition hypergraph CH(D) of a digraph D is the hypergraph such that the vertex set is the same as D and e ⊆ V (D) is a hyperedge if and only if e contains at least 2 vertices and e coincides with the in-neighborhood of some vertex v in the digraph D. Any hypergraph H with sufficiently many isolated vertices is the competition hypergraph of an acyclic digraph. The hypercompetition number...

One of the most interesting new developments in hypergraph colourings in these last few years has been Voloshin’s notion of colourings of mixed hypergraphs. In this paper we shall study a specific instance of Voloshin’s idea: a non-monochromatic non-rainbow (NMNR) colouring of a hypergraph is a colouring of its vertices such that every edge has at least two vertices coloured with different colo...

We generalize the notion of an Euler tour in a graph in the following way. An Euler family in a hypergraph is a family of closed walks that jointly traverse each edge of the hypergraph exactly once. An Euler tour thus corresponds to an Euler family with a single component. We provide necessary and sufficient conditions for the existence of an Euler family in an arbitrary hypergraph, and in part...

The talk deals with combinatorial problems concerning colorings of non-uniform hyper-graphs. Let H = (V, E) be a hypergraph with minimum edge-cardinality n. We show that if H is a simple hypergraph (i.e. every two distinct edges have at most one common vertex) and e∈E r 1−|e| c √ n, for some absolute constant c > 0, then H is r-colorable. We also obtain a stronger result for triangle-free simpl...

A hypergraph is just a set system: there is some ground set V , and the hypergraph is just some collection of subsets, F ⊆ 2V . The goal is to color the elements of V red and blue so that no element of F is monochromatic. Let’s assume that the hypergraph is k-uniform, namely that all sets in F have cardinality k. One natural approach is to color randomly: we independently assign each vertex the...