Publication

Linearity of saturation for Berge hypergraphs

Abhishek Methuku
2019
Journal Articles

Abstract

For a graph F, we say a hypergraph H is a Berge-F if it can be obtained from F by replacing each edge of F with a hyperedge containing it. We say a hypergraph is Berge-F-saturated if it does not contain a Berge-F, but adding any hyperedge creates a copy of a Berge-F. The k-uniform saturation number of Berge-F, sat(k)(n, Berge-F) is the fewest number of hyperedges in a Berge-F-saturated k-uniform hypergraph on n vertices. We show that sat(k)(n, Berge-F) = O(n) for all graphs F and uniformities 3

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Abhishek Methuku
2019
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/f0f6b941-f4ef-420e-be4e-a301b4ebe8ba

About this result

Related concepts (9)

Hypergraph

In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, a directed hypergraph is a pair , where is a set of elements called nodes, vertices, points, or elements and is a set of pairs of subsets of . Each of these pairs is called an edge or hyperedge; the vertex subset is known as its tail or domain, and as its head or codomain. The order of a hypergraph is the number of vertices in .

Matching in hypergraphs

In graph theory, a matching in a hypergraph is a set of hyperedges, in which every two hyperedges are disjoint. It is an extension of the notion of matching in a graph. Recall that a hypergraph H is a pair (V, E), where V is a set of vertices and E is a set of subsets of V called hyperedges. Each hyperedge may contain one or more vertices. A matching in H is a subset M of E, such that every two hyperedges e_1 and e_2 in M have an empty intersection (have no vertex in common).

Bipartite hypergraph

In graph theory, the term bipartite hypergraph describes several related classes of hypergraphs, all of which are natural generalizations of a bipartite graph. Property B The weakest definition of bipartiteness is also called 2-colorability. A hypergraph H = (V, E) is called 2-colorable if its vertex set V can be partitioned into two sets, X and Y, such that each hyperedge meets both X and Y. Equivalently, the vertices of H can be 2-colored so that no hyperedge is monochromatic.

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