Related papers: Minimum Degree Threshold for $H$-factors with High…

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…

For any bipartite graph $H$, we determine a minimum degree threshold for a balanced bipartite graph $G$ to contain a perfect $H$-tiling. We show that this threshold is best possible up to a constant depending only on $H$. Additionally, we…

A typical Dirac-type problem in extremal graph theory is to determine the minimum degree threshold for a graph $G$ to have a spanning subgraph $H$, e.g. the Dirac theorem. A natural following up problem would be to seek an $H$-factor, which…

A perfect $K_r$-tiling in a graph $G$ is a collection of vertex-disjoint copies of the clique $K_r$ in $G$ covering every vertex of $G$. The famous Hajnal--Szemer\'edi theorem determines the minimum degree threshold for forcing a perfect…

A proper $k$-colouring of a graph $G$ is called $h$-conflict-free if every vertex $v$ has at least $\min\, \{h, {\rm deg}(v)\}$ colours appearing exactly once in its neighbourhood. Let $\chi_{\rm pcf}^h(G)$ denote the minimum $k$ such that…

Given hypergraphs $F$ and $H$, an $F$-factor in $H$ is a set of vertex-disjoint copies of $F$ which cover all the vertices in $H$. Let $K^- _4$ denote the $3$-uniform hypergraph with $4$ vertices and $3$ edges. We show that for sufficiently…

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…

In this paper, we determine the minimum degree threshold of perfect matchings with high discrepancy in $r$-edge-colored $k$-uniform hypergraphs for all $k\geq 3$ and $r\geq 2$, thereby completing the investigation into discrepancies of…

For a proper vertex coloring $c$ of a graph $G$, let $\varphi_c(G)$ denote the maximum, over all induced subgraphs $H$ of $G$, the difference between the chromatic number $\chi(H)$ and the number of colors used by $c$ to color $H$. We…

The chromatic threshold $\delta_\chi(H)$ of a graph $H$ is the infimum of $d>0$ such that the chromatic number of every $n$-vertex $H$-free graph with minimum degree at least $dn$ is bounded in terms of $H$ and $d$. A breakthrough result of…

A fundamental result of K\"uhn and Osthus [The minimum degree threshold for perfect graph packings, Combinatorica, 2009] determines up to an additive constant the minimum degree threshold that forces a graph to contain a perfect H-tiling.…

A result of Balogh, Csaba, Jing and Pluh\'ar yields the minimum degree threshold that ensures a $2$-coloured graph contains a perfect matching of significant colour-bias (i.e., a perfect matching that contains significantly more than half…

For graphs $F$ and $H$, we say $F$ is Ramsey for $H$ if every $2$-coloring of the edges of $F$ contains a monochromatic copy of $H$. The graph $F$ is Ramsey $H$-minimal if $F$ is Ramsey for $H$ and there is no proper subgraph $F'$ of $F$ so…

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…

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 $G$ is called minimal if for any edge $e\in…

A graph $G$ is $k$-factor-critical if $G-S$ has a perfect matching for every subset $S \subseteq V(G)$ with $|S|=k$. A spanning subgraph $H$ of $G$ is called a $[1,b]$-odd factor if $b \equiv 1 \pmod{2}$ and $d_{H}(v) \in\left\lbrace 1, 3,…

We give, for each $k \geq 3$, the precise best possible minimum positive codegree condition for a perfect matching in a large $k$-uniform hypergraph $H$ on $n$ vertices. Specifically we show that, if $n$ is sufficiently large and divisible…

For a connected graph $G$, let $\mu(G)$ denote the distance spectral radius of $G$. A matching in a graph $G$ is a set of disjoint edges of $G$. The maximum size of a matching in $G$ is called the matching number of $G$, denoted by…

A proper vertex colouring of a graph $G$ is referred to as conflict-free if in the neighbourhood of every vertex some colour appears exactly once, while it is called $h$-conflict-free if there are at least $h$ such colours for each vertex…

The chromatic threshold delta_chi(H) of a graph H is the infimum of d>0 such that there exists C=C(H,d) for which every H-free graph G with minimum degree at least d|G| satisfies chi(G)<C. We prove that delta_chi(H) \in {(r-3)/(r-2),…