Related papers: Minimum codegree threshold for $C_6^3$-factors in …

Let $H$ and $F$ be hypergraphs. We say $H$ contains $F$ as a trace if there exists some set $S \subseteq V(H)$ such that $H|_S:=\{E\cap S: E \in E(H)\}$ contains a subhypergraph isomorphic to $F$. In this paper we give an upper bound on the…

Let $G$ be an edge-colored graph with $n$ vertices. A subgraph $H$ of $G$ is called a rainbow subgraph of $G$ if the colors of each pair of the edges in $E(H)$ are distinct. We define the minimum color degree of $G$ to be the smallest…

Let $G$ be an $n$-vertex graph and let $L:V(G)\rightarrow P(\{1,2,3\})$ be a list assignment over the vertices of $G$, where each vertex with list of size 3 and of degree at most 5 has at least three neighbors with lists of size 2. We can…

An $r$-uniform hypergraph is uniquely $k$-colorable if there exists exactly one partition of its vertex set into $k$ parts such that every edge contains at most one vertex from each part. For integers $k \ge r \ge 2$, let $\Phi_{k,r}$…

In this paper, we study the maximum number of edges in an $N$-vertex $r$-uniform hypergraph with girth $g$ where $g \in \{5,6 \}$. Writing $\textrm{ex}_r ( N, \mathcal{C}_{<g} )$ for this maximum, it is shown that $\textrm{ex}_r ( N ,…

In this paper, we prove the conjectures of Gharakhloo and Welker (2023) that the positive matching decomposition number (pmd) of a $3$-uniform hypergraph is bounded from above by a polynomial of degree $2$ in terms of the number of…

The celebrated Andr\'{a}sfai--Erd\H{o}s--S\'{o}s Theorem from 1974 shows that every $n$-vertex triangle-free graph with minimum degree greater than $2n/5$ must be bipartite. We establish a positive codegree extension of this result for the…

A subset $M$ of the edges of a graph or hypergraph is hitting if $M$ covers each vertex of $H$ at least once, and $M$ is $t$-shallow if it covers each vertex of $H$ at most $t$ times. We consider the existence of shallow hitting edge sets…

A graph $G$ of order $n$ is said to be $k$-factor-critical for integers $1\leq k < n$, if the removal of any $k$ vertices results in a graph with a perfect matching. $1$- and $2$-factor-critical graphs are the well-known factor-critical and…

Let $S=\{n_1,n_2,...,n_t\}$ be a finite set of positive integers with $\min(S)\geq 3$ and $t\geq 2$. For any positive integers $s_1,s_2,...,s_t$, we construct a family of 3-uniform bi-hypergraphs ${\cal H}$ with the feasible set $S$ and…

Let $F = (U,E)$ be a graph and $\mathcal{H} = (V,\mathcal{E})$ be a hypergraph. We say that $\mathcal{H}$ contains a Berge-$F$ if there exist injections $\psi:U\to V$ and $\varphi:E\to \mathcal{E}$ such that for every $e=\{u,v\}\in E$,…

We prove that every graph of minimum degree at least $d \ge 1$ contains a subdivision of some maximal 3-degenerate graph of order $d+1$. This generalizes the classic results of Dirac ($d=3$) and Pelik\'an ($d=4$). We conjecture that for any…

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…

For a $k$-uniform hypergraph (or simply $k$-graph) $F$, the codegree Tur\'{a}n density $\pi_{\mathrm{co}}(F)$ is the infimum over all $\alpha$ such that any $n$-vertex $k$-graph $H$ with every $(k-1)$-subset of $V(H)$ contained in at least…

Motivated by the work of Razborov about the minimal density of triangles in graphs we study the minimal density of the 5-cycle $C_5$. We show that every graph of order $n$ and size $\left( 1-\frac{1}{k}\right)\binom{n}{2}$, where $k\ge 3$…

The study of asymptotic minimum degree thresholds that force matchings and tilings in hypergraphs is a lively area of research in combinatorics. A key breakthrough in this area was a result of H\`{a}n, Person and Schacht who proved that the…

Let $m(n,r)$ denote the minimal number of edges in an $n$-uniform hypergraph which is not $r$-colorable. For the broad history of the problem see [RaiSh]. It is known that for a fixed $n$ the sequence \[ \frac{m(n,r)}{r^n} \] has a limit.…

Fix $r \ge 2$ and a collection of $r$-uniform hypergraphs $\cH$. What is the minimum number of edges in an $\cH$-free $r$-uniform hypergraph with chromatic number greater than $k$. We investigate this question for various $\cH$. Our results…

In 1968, Ringel and Youngs confirmed the last open case of the Heawood Conjecture by determining the genus of every complete graph $K_n$. In this paper, we investigate the minimum genus embeddings of the complete $3$-uniform hypergraphs…

Sidorenko's conjecture states that, for all bipartite graphs $H$, quasirandom graphs contain asymptotically the minimum number of copies of $H$ taken over all graphs with the same order and edge density. While still open for graphs, the…