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

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 $e=\{u,v\}\in E$,…

Let $F$ and $H$ be $k$-uniform hypergraphs. We say $H$ is $F$-saturated if $H$ does not contain a subgraph isomorphic to $F$, but $H+e$ does for any hyperedge $e\not\in E(H)$. The saturation number of $F$, denoted $\mathrm{sat}_k(n,F)$, is…

A fractional matching of $G$ is a function $f: E(G)\to [0,1]$ such that $\sum_{e\in E_G(v_i)}f(e)\le 1$ for any $v_i\in V(G)$, where $E_G(v_i)=\{e: e\in E(G) \ \textrm{and}\ e \ \textrm{is incident with} \ v_i\}$. Let $\alpha_f(G)$ denote…

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…

The \emph{minimum positive co-degree} of a non-empty $r$-graph ${H}$, denoted $\delta_{r-1}^+( {H})$, is the maximum $k$ such that if $S$ is an $(r-1)$-set contained in a hyperedge of $ {H}$, then $S$ is contained in at least $k$ distinct…

In this paper it is established that a decomposition of a 3-uniform hypergraph K_v^{(3)} into a special kind of hypergraph K_4^{(3)}+e exists if and only if v\equiv 0,1,2 (mod 5) and v\geq 7.

Let $G$ be a connected graph on $n$ vertices and at most $n(1+\epsilon)$ edges with bounded maximum degree, and $F$ a graph on $n$ vertices with minimum degree at least $n-k$, where $\epsilon$ is a constant depending on $k$. In this paper,…

The first part of the paper studies star-cycle factors of graphs. It characterizes star-cycle factors of a graph $G$ and proves upper bounds for the minimum number of $K_{1,2}$-components in a $\{K_{1,1}, K_{1,2}, C_n\colon n\ge 3\}$-factor…

Given two $k$-graphs ($k$-uniform hypergraphs) $F$ and $H$, a perfect $F$-tiling (or an $F$-factor) in $H$ is a set of vertex disjoint copies of $F$ that together cover the vertex set of $H$. For all complete $k$-partite $k$-graphs $K$,…

We investigate extremal problems for hypergraphs satisfying the following density condition. A $3$-uniform hypergraph $H=(V, E)$ is $(d, \eta,P_2)$-dense if for any two subsets of pairs $P$, $Q\subseteq V\times V$ the number of pairs…

A spanning subgraph $F$ of a graph $G$ is called a $[1,b]$-odd factor if $b\equiv1$ (mod 2) and $d_F(v)\in\{1,3,\ldots,b\}$ for every $v\in V(G)$. A graph $G$ of order $n\geq k+2$ is $k$-critical with respect to $[1,b]$-odd factor if for…

We first prove that for every vertex x of a 4-connected graph G there exists a subgraph H in G isomorphic to a subdivision of the complete graph K4 on four vertices such that G-V(H) is connected and contains x. This implies an affirmative…

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. A $k$-factor-critical graph is minimal if for every edge, the deletion of…

An even factor of $G$ is a spanning subgraph $F$ such that every vertex in $F$ has a nonzero even degree. Note that $\delta(G)\geq2$ is a trivial necessary condition for a graph to have an even factor, where $\delta(G)$ is the minimum…

We prove that every $n$-vertex 4-uniform hypergraph with minimum codegree at least $\lfloor n/4 \rfloor$ has a spanning tight component. This is tight, and it settles the 4-uniform case of a conjecture of Illingworth, Lang, M\"uyesser,…

We determine, up to a multiplicative constant, the optimal number of random edges that need to be added to a $k$-graph $H$ with minimum vertex degree $\Omega(n^{k-1})$ to ensure an $F$-factor with high probability, for any $F$ that belongs…

A perfect $K_t$-matching in a graph $G$ is a spanning subgraph consisting of vertex disjoint copies of $K_t$. A classic theorem of Hajnal and Szemer\'edi states that if $G$ is a graph of order $n$ with minimum degree $\delta(G) \ge…

Given a graph $F$, let $I(F)$ be the class of graphs containing $F$ as an induced subgraph. Let $W[F]$ denote the minimum $k$ such that $I(F)$ is definable in $k$-variable first-order logic. The recognition problem of $I(F)$, known as…

Let $\mathcal{H}$ be an $r$-uniform hypergraph and $F$ be a graph. We say $\mathcal{H}$ contains $F$ as a trace if there exists some set $S\subseteq V(\mathcal{H})$ such that $\mathcal{H}|_{S}:=\{E\cap S: E\in E(\mathcal{H})\}$ contains a…

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…