Related papers: Localized Version of Hypergraph Erdos-Gallai Theor…

In this paper, we prove a theorem on tight paths in convex geometric hypergraphs, which is asymptotically sharp in infinitely many cases. Our geometric theorem is a common generalization of early results of Hopf and Pannwitz [12],…

For fixed integers $r\ge 3,e\ge 3,v\ge r+1$, an $r$-uniform hypergraph is called $\mathscr{G}_r(v,e)$-free if the union of any $e$ distinct edges contains at least $v+1$ vertices. Brown, Erd\H{o}s and S\'{o}s showed that the maximum number…

For two graphs $F$ and $H$, the relative Tur\'{a}n number $\mathrm{ex}(H,F)$ is the maximum number of edges in an $F$-free subgraph of $H$. Foucaud, Krivelevich, and Perarnau \cite{FKP} and Perarnau and Reed \cite{PR} studied these…

Let $H=(V,E)$ be an $r$-uniform hypergraph with the vertex set $V$ and the edge set $E$. For $1\leq s \leq r/2$, we define a weighted graph $G^{(s)}$ on the vertex set ${V\choose s}$ as follows. Every pair of $s$-sets $I$ and $J$ is…

The Erd\H{o}s-S\'os Conjecture states that every graph with average degree exceeding $k-1$ contains every tree with $k$ edges as a subgraph. We prove that there are $\delta>0$ and $k_0\in\mathbb N$ such that the conjecture holds for every…

For a graph $F$, we say a hypergraph $H$ is Berge-$F$ if it can be obtained from $F$ be 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…

A famous conjecture of Erd\H{o}s and S\'os states that every graph with average degree more than $k - 1$ contains all trees with $k$ edges as subgraphs. We prove that the Erd\H{o}s-S\'os conjecture holds approximately, if the size of the…

An old result of M\"uller and R\"odl states that a countable graph $G$ has a subgraph whose vertices all have infinite degree if and only if for any vertex labeling of $G$ by positive integers, an infinite increasing path can be found. They…

The Harary-Sachs theorem for $k$-uniform hypergraphs equates the codegree-$d$ coefficient of the adjacency characteristic polynomial of a uniform hypergraph with a weighted sum of subgraph counts over certain multi-hypergraphs with $d$…

Let $H=(V,E)$ be a hypergraph, where $V$ is a set of vertices and $E$ is a set of non-empty subsets of $V$ called edges. If all edges of $H$ have the same cardinality $r$, then $H$ is a $r$-uniform hypergraph; if $E$ consists of all…

For every $r\in \mathbb{N}$, we denote by $\theta_{r}$ the multigraph with two vertices and $r$ parallel edges. Given a graph $G$, we say that a subgraph $H$ of $G$ is a model of $\theta_{r}$ in $G$ if $H$ contains $\theta_{r}$ as a…

The Erd\H{o}s--Hajnal Theorem asserts that non-universal graphs, that is, graphs that do not contain an induced copy of some fixed graph $H$, have homogeneous sets of size significantly larger than one can generally expect to find in a…

Given a graph $F$, a Berge copy of $F$ (Berge-$F$ for short) is a hypergraph obtained by enlarging the edges arbitrarily. Gy\H{o}ri, Salia and Zamora determined the maximum number of hyperedges in a connected $r$-uniform hypergraph on $n$…

A graph or hypergraph is said to be vertex-transitive if its automorphism group acts transitively upon its vertices. A classic theorem of Mader asserts that every connected vertex-transitive graph is maximally edge-connected. We generalise…

For every natural number $p$, we show that the maximum negative girth among the class of signed graphs on $n$ vertices with balanced chromatic number at least $p$ is between $(1/e)n^{1/(p-1)}$ and $2(p-1) n^{1/(p-1)}$. This extends a…

The problem of finding the maximum-weight, planar subgraph of a finite, simple graph with nonnegative real edge weights is well known in industrial and electrical engineering, systems biology, sociology and finance. As the problem is known…

Given a graph $F$, a hypergraph is a Berge-$F$ if it can be obtained by expanding each edge in $F$ to a hyperedge containing it. A hypergraph $H$ is Berge-$F$-saturated if $H$ does not contain a subgraph that is a Berge-$F$, but for any…

For a graph G, a hypergraph H is called Berge-G if there is a hypergraph H', isomorphic to H, containing all vertices of G, so that e is contained in f(e) for each edge e of G, where f is a bijection between E(G) and E(H'). The set of all…

In this paper we introduce the concept of minimum-weight edge-discriminators in hypergraphs, and study its various properties. For a hypergraph $\mathcal H=(\mathcal V, \mathcal E)$, a function $\lambda: \mathcal V\rightarrow \mathbb…

We consider the Erd\H{o}s-R\'enyi evolution of random graphs, where a new uniformly distributed edge is added to the graph in every step. For every fixed $d\ge 1$, we show that with high probability, the graph becomes rigid in $\mathbb R^d$…