Related papers: On asymmetric hypergraphs

In this paper, we prove that for any $k\ge 3$, there exist infinitely many minimal asymmetric $k$-uniform hypergraphs. This is in a striking contrast to $k=2$, where it has been proved recently that there are exactly $18$ minimal asymmetric…

A $k$-graph $\mathcal{G}$ is asymmetric if there does not exist an automorphism on $\mathcal{G}$ other than the identity, and $\mathcal{G}$ is called minimal asymmetric if it is asymmetric but every non-trivial induced sub-hypergraph of…

Confirming a conjecture of Ne\v{s}et\v{r}il, we show that up to isomorphism there is only a finite number of finite minimal asymmetric undirected graphs. In fact, there are exactly 18 such graphs. We also show that these graphs are exactly…

In this note, we determine the maximum number of edges of a $k$-uniform hypergraph, $k\ge 3$, with a unique perfect matching. This settles a conjecture proposed by Snevily.

We prove that for any $k \ge 3$, every $k$-uniform hypergraph on $n$ vertices contains at most $n - \omega(1)$ different sizes of cliques (maximal complete subgraphs). In particular, the 3-uniform case answers a question of Erd\H{o}s.

In this paper, we introduce super-minimally $k$-connected graphs, those $k$-connected graphs in which no proper subgraph is $k$-connected. For $k$ greater than or equal to three, this class lies strictly between the classes of minimally…

A $k$-uniform hypergraph $M$ is set-homogeneous if it is countable (possibly finite) and whenever two finite induced subhypergraphs $U,V$ are isomorphic there is $g\in Aut(M)$ with $U^g=V$; the hypergraph $M$ is said to be homogeneous if in…

For any given integer $r\geqslant 3$, let $k=k(n)$ be an integer with $r\leqslant k\leqslant n$. A hypergraph is $r$-uniform if each edge is a set of $r$ vertices, and is said to be linear if two edges intersect in at most one vertex. Let…

A $k$-uniform hypergraph is $s$-almost intersecting if every edge is disjoint from exactly $s$ other edges. Gerbner, Lemons, Palmer, Patk\'os and Sz\'ecsi conjectured that for every $k$, and $s>s_0(k)$, every $k$-uniform $s$-almost…

The main result of this paper is that for any $c>0$ and for large enough $n$ if the number of edges in a 3-uniform hypergraph is at least $cn^2$ then there is a core (subgraph with minimum degree at least 2) on at most 15 vertices. We…

In this article we investigate the structure of uniformly $k$-connected and uniformly $k$-edge-connected graphs. Whereas both types have previously been studied independent of each other, we analyze relations between these two classes. We…

We prove that for every integer $r\geq 2$, an $n$-vertex $k$-uniform hypergraph $H$ containing no $r$-regular subgraphs has at most $(1+o(1)){{n-1}\choose{k-1}}$ edges if $k\geq r+1$ and $n$ is sufficiently large. Moreover, if…

We call a finite undirected graph minimally k-matchable if it has at least k distinct perfect matchings but deleting any edge results in a graph which has not. An odd subdivision of some graph G is any graph obtained by replacing every edge…

Let $H$ be a $k$-uniform hypergraph on $n$ vertices where $n$ is a sufficiently large integer not divisible by $k$. We prove that if the minimum $(k-1)$-degree of $H$ is at least $\lfloor n/k \rfloor$, then $H$ contains a matching with…

We prove an asymptotic formula for the number of $k$-uniform hypergraphs with a given degree sequence, for a wide range of parameters. In particular, we find a formula that is asymptotically equal to the number of $d$-regular $k$-uniform…

Let $k\ge 3$ be an odd integer and let $n$ be a sufficiently large integer. We prove that the maximum number of edges in an $n$-vertex $k$-uniform hypergraph containing no $2$-regular subgraphs is $\binom{n-1}{k-1} + \lfloor\frac{n-1}{k}…

In this article we give an explicit classification for the countably infinite graphs $\mathcal{G}$ which are, for some $k$, $\geq$$ k$-homogeneous. It turns out that a $\geq$$k-$homogeneous graph $\mathcal{M}$ is non-homogeneous if and only…

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…

The transversal number $\tau(H)$ of a hypergraph $H$ is the minimum number of vertices that intersect every edge of $H$. A linear hypergraph is one in which every two distinct edges intersect in at most one vertex. A $k$-uniform hypergraph…

The size-Ramsey number $R^{(k)}(H)$ of a $k$-uniform hypergraph $H$ is the minimum number of edges in a $k$-uniform hypergraph $G$ with the property that every `$2$-edge coloring' of $G$ contains a monochromatic copy of $H$. For $k\ge2$ and…