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# Questions tagged [hypergraph]

Hypergraphs are generalizations of graphs, where edges can be made of more than two vertices.

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### The vertex-covering number of a particular hypergraph

$\newcommand{\cM}{{\mathcal M}}$ For an integer $n>0$, let $\cM_n$ denote the set of all matrices with three rows and $n$ columns such that every column is obtained by permitting the coordinates ...
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### Name for a specific kind of regular hyper graphs

Question: is there already an established name for the following kind of hypergaphs: given a set $\mathfrak{V}$ with $n\lt\infty$ elements the hyper vertices $\mathfrak{v}\subseteq \mathfrak{V}$ are ...
1 vote
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### Is the chromatic number of hypergraphs downward continuous?

Let $H=(V,E)$ be a hypergraph. The chromatic number $\chi(H)$ is the smallest cardinal $\kappa \leq |V|$ such that there is a map $c:V\to \kappa$ such that for all $e\in E$ having more than one ...
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### Maximum number of h-dimensional hyper-edges without forming any (h+1) complete subgraph

This is about graph theory. Define an h-dimensional hyperedge as a set that contains h vertices. A graph of (h+1) vertices is h-complete if any h combination (or any subset with size h) is an h-...
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### Number of triangle-free graphs with prescribed number of edges

This question is posted from StackExchange since it received no answer there. Let $f(n, e)$ be the number of triangle-free graphs on $n$ vertices and $e$ edges. From empirical evidence, I am motivated ...
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### Visiting zero-sum triples in a vector space

Is it true that for any set $A\subset\mathbb F_5^n$ satisfying $A\cap(-A)=\varnothing$, there is a subset $A'\subseteq A$ such any triple $(a,b,c)\in A\times A\times A$ with $a+b+c=0$ has exactly one ...
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### Line graphs of complete hypergraphs as complement of Kneser graphs

Since the Johnson graph/triangular graph $J(n,2)$ is the complement of the Kneser graph $K(n,2)$, which is also incidentally the line graph of the complete graph $K_n$, I thought whether the same can ...
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### Can we talk about approximation when the decision problem for solution existence is NP-Hard

I am wishing to design an approximation algorithm for an optimization problem where the existence of solution for corresponding decision problem is not guaranteed. Is it wise to find an approximation ...
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### Choice sets in hypergraphs with finite edges

Let $H=(V,E)$ be a hypergraph. We say that $C\subseteq V$ is a choice set if $|C\cap e| = 1$ for all $e\in E$. Question. Let $H=(V,E)$ be a hypergraph with $e$ finite for all $e\in E$, and suppose ...
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### Hypergraphs such that all finite subhypergraphs are bipartite

The starting point of this question is the following true statement for graphs: A simple, undirected graph $G = (V,E)$ is bipartite if and only if for all $E_0\subseteq E$ the graph $(V, E_0)$ is ...
1 vote
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### Hypergraphs with finite matching / covering balance

Let $H=(V,E)$ be a hypergraph such that $\emptyset\notin E$. We say that $C\subseteq V$ is a (vertex) cover if for all $e \in E$ we have $C\cap e\neq \emptyset$. The minimum size that a cover can have ...
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### Discrepancy of random bipartite graphs (2)

This question is a modification of the one asked here, which turned out to ask for something too strong to be true. Given $k>0$ and a positive integer $n$, let $X, Y$ be two vertex sets of size $n$ ...
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### Discrepancy of random bipartite graphs

This is a crosspost from MathStackExchange (original question). Fix $k>0$ and let $X, Y$ be two vertex sets of size $n$ a positive integer (we're interested in the limit $n\to \infty$). Define a ...
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### Edge sets on $\omega$ maximal with respect to chromatic number

If $H=(V,E)$ is a hypergraph, and $\kappa \neq \emptyset$ is a cardinal, a map $c: V\to \kappa$ is said to be a coloring if the restriction $c|_e: e\to \kappa$ is non-constant whenever $e\in E$ and $e$...
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### Conjecture on connected hypergraphs

A hypergraph $H=(V,E)$ with $V$ non-empty is said to be connected if for all $S\subseteq V$ with $\emptyset \neq S \neq V$ there is $e\in E$ such that $e$ intersects both $S$ and $V\setminus S$. Given ...
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### Conflict-free coloring of linear hypergraphs on $\omega$

This question is motivated by considerations on conflict-free colorings, which arose while studying assignment problems for frequencies in cellular networks. A hypergraph $H=(V,E)$ is said to be ...
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### A form of Hadwiger's conjecture for hypergraphs

A hypergraph $H=(V,E)$ consists of a set $V$ and $E\subseteq {\mathcal P}(V)$. If $S\subseteq V$, we define $$E|_S = \{e\cap S: (e\in E) \land (e\cap S \neq \emptyset)\}$$ and call $(S, E|_S)$ the ...
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### Chromatic number of duals of uniform hypergraphs with large edges

Let $H=(V,E)$ be a hypergraph. If $\kappa>0$ is a cardinal, we say that $H$ is $\kappa$-uniform if $|e|=\kappa$ for all $e\in E$. If $X$ is a non-empty set, then a map $c:V\to X$ is said to be a ...
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### Matching number in infinite hypergraphs

If $H= (V,E)$ is a hypergraph, a matching is a set $M\subseteq E$ such that $e_1\cap e_2 = \emptyset$ whenever $e_1\neq e_2 \in M$. The matching number $\mu(H)$ of a hypergraph $H=(V,E)$ with $V$ ...
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### Chromatic index of hypergraphs

A proper $k$-edge-coloring of a hypergraph $H$ is a mapping from $E(H)$ to a set of $k$ colors so that every pair of adjacent edges receives different colors. We say $H$ is $k$-edge-colorable if $H$ ...
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### 3-uniform tetrahedron-free hypergraph on seven vertices

My problem concerns 3-uniform hypergraphs. Let $f(n)$ be the maximal number of edges in a 3-uniform hypergraph such that no four edges form a "tetrahedron", i.e., four edges that join the ...
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### Are strongly complete regular linear hypergraphs on $\omega$ isomorphic?

This is a related question to an older one. If $H_i = (V_i, E_i)$ are hypergraphs for $i=1,2$ then we say they are isomorphic if there is a bijection $f: V_1 \to V_2$ such that for $A \subseteq V_1$ ...
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### Cardinalities of maximal linear $k$-subsets of $n = \{0,\ldots,n-1\}$

We consider any non-negative integer $n$ as a cardinal, so $0 = \emptyset$, and $n=\{0,\ldots,n-1\}$ for positive $n$. Given $n,k\in \mathbb{N}$, let $[n]^k$ denote the collection of $k$-element ...
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### Chromatic number of $(n, [n]^k)$

If $n\in\mathbb{N}$ is a non-negative integer, we consider it as a cardinal, so $n = \{0, \ldots, n-1\}$. If $X$ is a set, and $\kappa$ is a cardinal, we let $[X]^\kappa$ be the collection of subsets ...
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### Self-dual hypergraph on $\omega$

Let $H=(V,E)$ be a hypergraph. For $v\in V$ we let $v^* = \{e\in E:v\in e\}$. We define the dual of $H$ by $H^*= (E, V^*)$ where $V^* = \{v^*: v\in V\}$. We say that a $H$ is self-dual if $H \cong H^*$...
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### Number of nonisomorphic weighted hypergraphs of certain type

Let $G=(V,E)$ be an unlabeled simple hypergraph with weighted vertices and given properties: $|v|⩾max(d(v);\;3)\;∀v∈V$, where $|v|$ denotes weight of vertice $v$ and $d(v):=\#(e:\;v∈e)$ - number of ...
If $H=(V,E)$ is a hypergraph and $\kappa \neq \emptyset$ is a cardinal, then a map $c:V\to\kappa$ is called a coloring if the restriction $c\restriction_e$ is non-constant for all $e\in E$ with $|e|\... 2 votes 1 answer 45 views ### Summable hypergraphs A hypergraph$H=(V,E)$is sane if$V$is finite,$E \neq \emptyset$, and$\emptyset \notin E$, and$e\not\subseteq e'$whenever$e\neq e' \in E$. Moreover, we call$H$summable if there is a map$f:V\...
Let $H=(V,E)$ be a hypergraph. If $\kappa \neq \emptyset$ is a cardinal, we call a map $c:V\to \kappa$ a coloring if for each $e\in E$ with $|e|>1$ the restriction $c\restriction_e$ is non-constant....