Related papers: Minimum codegree threshold for $(K_4^3-e)$-factors

A result of Gy\'arf\'as says that for every $3$-coloring of the edges of the complete graph $K_n$, there is a monochromatic component of order at least $\frac{n}{2}$, and this is best possible when $4$ divides $n$. Furthermore, for all…

Given a family of 3-graphs F its codegree threshold coex(n, F) is the largest number d=d(n) such that there exists an n-vertex 3-graph in which every pair of vertices is contained in at least d 3-edges but which contains no member of F as a…

Let $G=(V,E)$ be a graph of density $p$ on $n$ vertices. Following Erd\H{o}s, \L uczak and Spencer, an $m$-vertex subgraph $H$ of $G$ is called {\em full} if $H$ has minimum degree at least $p(m - 1)$. Let $f(G)$ denote the order of a…

We investigate a covering problem in $3$-uniform hypergraphs ($3$-graphs): given a $3$-graph $F$, what is $c_1(n,F)$, the least integer $d$ such that if $G$ is an $n$-vertex $3$-graph with minimum vertex degree $\delta_1(G)>d$ then every…

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…

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…

A fractional matching of a graph $G$ is a function $f:E(G)\rightarrow [0, 1]$ such that for any $v\in V(G)$, $\sum_{e\in E_{G}(v)}f(e)\leq1$, where $E_{G}(v)=\{e\in E(G): e~ \mbox{is incident with} ~v~\mbox{in}~G\}$.The fractional matching…

Given two $r$-uniform hypergraphs $F$ and $H$, we say that $H$ has an $F$-covering if every vertex in $H$ is contained in a copy of $F$. Let $c_{i}(n,F)$ be the least integer such that every $n$-vertex $r$-graph $H$ with…

We study minimum vertex-degree conditions in 3-uniform hypergraphs for (tight) spanning components and (combinatorial) surfaces. Our main results show that a 3-uniform hypergraph $G$ on $n$ vertices contains a spanning component if…

Let H be any graph. We determine (up to an additive constant) the minimum degree of a graph G which ensures that G has a perfect H-packing (also called an H-factor). More precisely, let delta(H,n) denote the smallest integer t such that…

Given $k\ge 2$ and two $k$-graphs ($k$-uniform hypergraphs) $F$ and $H$, an $F$-factor in $H$ is a set of vertex-disjoint copies of $F$ that together covers the vertex set of $H$. Lenz and Mubayi [J. Combin. Theory Ser. B, 2016] studied the…

Given a 2-edge-coloring $f : E(K_n) \rightarrow \{\pm 1\}$, the discrepancy of a subgraph $F \subseteq K_n$ is defined as $\left| \sum_{e \in E(F)} f(e) \right|$. Erd\H{o}s, F\"uredi, Loebl and S\'os showed that if $F$ is an $n$-vertex tree…

A pseudo [2,b]-factor of a graph G is a spanning subgraph in which each component C on at least three vertices is a [2,b]-graph. The main contibution of this paper, is to give an upper bound to the number of components that are edges or…

A classical result by Hajnal and Szemer\'edi from 1970 determines the minimal degree conditions necessary to guarantee for a graph to contain a $K_r$-factor. Namely, any graph on $n$ vertices, with minimum degree $\delta(G) \ge…

For a hypergraph $H=(V,\mathcal E)$, a subfamily $\mathcal C\subseteq \mathcal E$ is called a cover of the hypergraph if $\bigcup\mathcal C=\bigcup\mathcal E$. A cover $\mathcal C$ is called minimal if each cover $\mathcal…

For a hypergraph $\mathcal{H}$, define the minimum positive codegree $\delta_i^+(\mathcal{H})$ to be the largest integer $k$ such that every $i$-set which is contained in at least one edge of $\mathcal{H}$ is contained in at least $k$…

Let $G=(V(G),E(G)) $ be a graph with vertex set $V(G)$ and edge set $E(G)$. An even factor of $G$ is a spanning subgraph $F$ such that every vertex in $F$ has a nonzero even degree. Note that $\delta(G)\geq 2$ is a trivial necessary…

For a $k$-uniform hypergraph (or simply $k$-graph) $F$, the codegree Tur\'{a}n density $\pi_{\mathrm{co}}(F)$ is the supremum over all $\alpha$ such that there exist arbitrarily large $n$-vertex $F$-free $k$-graphs $H$ in which every…

Given a family of 3-graphs $F$, we define its codegree threshold $\mathrm{coex}(n, F)$ to be the largest number $d=d(n)$ such that there exists an $n$-vertex 3-graph in which every pair of vertices is contained in at least $d$ 3-edges but…

For graphs $F$ and $H$, let $f_{F,H}(n)$ be the minimum possible size of a maximum $F$-free induced subgraph in an $n$-vertex $H$-free graph. This notion generalizes the Ramsey function and the Erd\H{o}s--Rogers function. Establishing a…