We consider two natural generalizations of the notion of transversal to a finite hypergraph, arising in data-mining and machine learning, the so called multiple and partial transversals. We show that for an arbitrary hypergraph the families of multiple and partial transversals are both dual-bounded in the sense that the size of the corresponding dual hypergraph is bounded by a polynomial in the...

We prove a hypergraph analogue of the Erdős-Gallai theorem: for any δ > 0 there exists n0 such that for any 0 ≤ α ≤ 1, any k-uniform hypergraph H on n ≥ n0 vertices with at least (α+ δ) ( n k ) edges contains a tight cycle of length at least αn. Moreover, H contains tight cycles of all lengths up to αn which are divisible by k. A key tool in our proof is a ‘regular slice lemma’, which is derive...

The Füredi–Hajnal conjecture, originally posed in [6] and solved by Marcus and Tardos in [10], can be cast as an extremal problem on ordered graphs. It states that any n-vertex bipartite graph that avoids a fixed permutation graph has O(n) edges. Balogh, Bollobás and Morris [1] recently generalized this result to show that any n-vertex, ordered hypergraph that avoids a fixed permutation graph h...

The expansion of a hypergraph, a natural extension of the notion of expansion in graphs, is defined as the minimum over all cuts in the hypergraph of the ratio of the number of the hyperedges cut to the size of the smaller side of the cut. We study the Hypergraph Small Set Expansion problem, which, for a parameter δ ∈ (0, 1/2], asks to compute the cut having the least expansion while having at ...

Problems in scientific computing, such as distributing large sparse matrix operations, have analogous formulations hypergraph partitioning problems. A is a generalization of traditional graph wherein “hyperedges” may connect any number nodes. As result, an NP-Hard problem to both solve or approximate. State-of-the-art algorithms that this follow the multilevel paradigm, which begins by iterativ...

Hypergraphs are a generalization of graphs where edges (aka nets ) allowed to connect more than two vertices. They have similarly wide range applications as graphs. This article considers the fundamental and intensively studied problem balanced hypergraph partitioning (BHP) , which asks for vertices into k disjoint blocks bounded size while minimizing an objective function over hyperedges. Here...

Let σ be a partition of the positive integer r. A σ-hypergraph H = H(n, r, q|σ) is an r-uniform hypergraph on nq vertices which are partitioned into n classes V1, V2, . . . , Vn each containing q vertices. An r-subset K of vertices is an edge of the hypergraph if the partition of r formed by the non-zero cardinalities |K ∩ Vi|, 1 ≤ i ≤ n, is σ. In earlier works we have considered colourings of ...

A hypergraph is k-irregular if there is no set of k vertices all of which have the same degree. We asymptotically determine the probability that a random uniform hypergraph is k-irregular.

A mixed hypergraph consists of two families of edges: the C-edges and D-edges. In a coloring every C-edge has at least two vertices of the same color, while every D-edge has at least two vertices colored differently. The largest and smallest possible numbers of colors in a coloring are termed the upper and lower chromatic number, χ̄ and χ, respectively. A mixed hypergraph is called uniquely colo...

The capacity of a directed hypergraph is a particular numerical quantity associated with a hypergraph. It is of interest because of certain important connections to longstanding conjectures in theoretical computer science related to fast matrix multiplication and perfect hashing as well as various longstanding conjectures in extremal combinatorics. We give an overview of the concept of the capa...