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Questions tagged [hypergraphs]

Use this tag for questions about hypergraphs, i.e. generalizations of graphs in graph theory, in which edges are allowed to be arbitrary subsets of vertices, instead of just pairs.

Hypergraphs are a generalization of graphs in which an edge can join any number of vertices, instead of just two.

Formally, a hypergraph consists of:

  • a set of vertices;
  • a family of sets of vertices, called hyperedges or just edges.

We often consider $k$-uniform hypergraphs for some value of $k$: in these, all hyperedges contain exactly $k$ vertices. Thus, graphs are exactly the $2$-uniform hypergraphs.

One standard references for this area is Berge's Graphs and Hypergraphs.

171 questions
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Minimum Cake Cutting for a Party

You are organizing a party. However, the number of guests to attend your party can be anything from $a_1$, $a_2$, $\ldots$, $a_n$, where the $a_i$'s are positive integers. You want to be prepared, so you want to cut a cake into smaller pieces. …
Batominovski
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Coarse-graining a bounded-incidence hypergraph

Let $G$ be a connected hypergraph with vertex-set $V$ and hyperedge-set $E \subseteq 2^V\smallsetminus \{\emptyset\}$. Assume that every vertex is part of at most $\Delta$ hyperedges and every hyperedge has at most $\Delta$ vertices. I call such $G$…
Pranay
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Are hypergraphs more expressive than graphs?

I started studying hypergraphs theory some days ago. I know that a hypergraph is a tuple $H = (X, E)$, in which $E \subseteq \mathcal{P}(X)$ and is actually a generalisation of the notion of graph. Though, I'm wondering why they're useful. I saw…
danin
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Extending a theorem from bipartite graphs to tripartite hypergraphs

Here is a useful theorem on bipartite graphs. Theorem. Let $G = (X\cup Y,E)$ be a bipartite graph with $n$ vertices in each side and positive weights on the edges. If for each vertex $v \in X\cup Y$, the sum of weights of edges adjacent to $v$ is…
Erel Segal-Halevi
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Definition of Category of Hypergraphs

I have a question concerning the definition of Hypergraphs in category theory, which I adopted from "A category-theoretical approach to hypergraphs" by W.Dörfler and D.A.Waller: http://rd.springer.com/article/10.1007%2FBF01224952#page-1 So the…
Mari Gachi
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Hypergraph $2$-colorability is NP-complete

So far all my searches for a proof of this well-known theorem have led me to the one below (Lovász 1973), reducing $k$-colorability for ordinary graphs to $2$-colorability for hypergraphs. In the construction, $H$ seems to contain copies of $G$, so…
Samuel Coskey
  • 471
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Number of colours in chains of Kneser colouring

Consider the set $[n]=\{1,2,\dots, n\}$. A Kneser $k$-colouring of $[n]$ is a map $c:\mathcal P([n])\to S$ to some set $S$ such that for sets $A_1, A_2,\dots, A_k$ we have $c(A_i)=c(A_j)$ for all $i,j$ implies $\cap_{i=1}^kA_i\ne \varnothing$. Graph…
HackR
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Hypergraph variant of handshake problem.

I came across this well known problem that goes something like this. If $n$ people shake hands with each other. How many handshakes will be there in total? The question can be interpreted as asking about the number of edges in a complete graph of…
Bejofo
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Are there holes in my road map from calculus to malliavin differential geometry, Bayesian hypergraphs, and causal inference?

I've constructed a directed acyclic graph that leads from introductory subjects, such as calculus (single and multivariable) to some of my current interests including causal inference, Bayesian hypergraphs, and Malliavin differential geometry. The…
Galen
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Threshold probability of the 3-uniform hypertriangle

I am stuck on a random hypergraph problem (I am encountering random hypergraphs for the very first time). Let $G_{3}(n, p)$ be a binomial 3-uniform hypergraph. Find a threshold probability for containing a 3-uniform hypertriangle. What I have tried:…
SpeedForce
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three-uniform hypergraph on $n$ vertices with at least $n/3$ edges contains an independent set of size at least $\frac{2n^{3/2}}{3\sqrt{3m}}$

Here is question 3 from chapter 3 Part 1 of The Probabilistic Method, 4th edition. Prove that every three-uniform hypergraph with $n$ vertices and $m \ge n∕3$ edges contains an independent set (i.e., a set of vertices containing no edges) of size…
H. Walter
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Bound for the number of edges of a linear uniform hypergraph

A hypergraph $H$ is a pair $H = (X,E)$ where $X$ is a set of elements called nodes or vertices, and $E$ is a set of non-empty subsets of $X$ called hyperedges or edges. We say a hypergraph is $k$-uniform if all its hyperedges have size…
Santiago Armstrong
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Bruhat-Tits Building of $PGL_3$: What does it look like?

I am very new to this topic, and my background is mostly in graph theory and basic algebra. What I want for now is to understand the structure of the dimension $2$ complex $\mathcal{B}(PGL_3(K))$ where $K$ is a $p$-adic field. In this case, we have…
BharatRam
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Hypergraph terminology

Suppose I have a hypergraph with vertices V and hyperedges H, where each hyperedge is a subset of V. I want to form a normal graph with vertex set V, where two vertices are adjacent if they lie in the same hyperedge. What is this graph called? If…
Gordon Royle
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Does this property completely define SET-like games?

In the game SET each card has $4$ properties (color, filling, shape and number), each with $3$ possible values, for a total of $3 ^ 4 = 81$ cards. We can think of cards as elements of $\{0,1,2\} ^ 4$. 3 cards $x,y$ and $z$ are said to form a SET if…
Ynir Paz
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