相关论文: On the co-degree threshold for the Fano plane

We show that every $3$-connected $K_{2,\ell}$-minor free graph with minimum degree at least $4$ has maximum degree at most $7\ell$. As a consequence, we show that every 3-connected $K_{2,\ell}$-minor free graph with minimum degree at least…

For an ordered graph $F$, denote the Tur\'an density by $\vec{\pi}(F)$. The relative Tur\'an density, denoted by $\rho(F)$, is the supremum over $\alpha \in [0,1]$ such that every ordered graph $G$ contains an $F$-free subgraph $G'$ with…

In 2006, K\"{u}hn and Osthus showed that if a 3-graph H on n vertices has minimum co-degree at least (1/4 +o(1))n and n is even then H has a loose Hamilton cycle. In this paper, we prove that the minimum co-degree of n/4 suffices. The…

In 1941, Turan conjectured that the edge density of any 3-graph without independent sets on 4 vertices (Turan (3,4)-graph) is >= 4/9(1-o(1)), and he gave the first example witnessing this bound. Brown (1983) and Kostochka (1982) found many…

A 1-plane graph is a graph together with a drawing in the plane in such a way that each edge is crossed at most once. A 1-plane graph is maximal if no edge can be added without violating either 1-planarity or simplicity. Let $m(n)$ denote…

Given hypergraphs H and F, an F-factor in H is a spanning subgraph consisting of vertex disjoint copies of F. Let K_4^3-e denote the 3-uniform hypergraph on 4 vertices with 3 edges. We show that for \gamma>0 there exists an integer n_0 such…

Let $Y_{3,2}$ be the 3-graph with two edges intersecting in two vertices. We prove that every 3-graph $ H $ on $ n $ vertices with at least $ \max \left \{ \binom{4\alpha n}{3}, \binom{n}{3}-\binom{n-\alpha n}{3} \right \}+o(n^3) $ edges…

A graph of order $n$ is said to be $k$-\emph{factor-critical} $(0\le k<n)$ if the removal of any $k$ vertices results in a graph with a perfect matching. A $k$-factor-critical graph $G$ is \emph{minimal} if $G-e$ is not $k$-factor-critical…

Let $C_6^3$ be the 3-uniform hypergraph on $\{1,\dots, 6\}$ with edges $123, 345,561$, which can be seen as the triangle in 3-uniform hypergraphs. For sufficiently large $n$ divisible by 6, we show that every $n$-vertex 3-uniform hypergraph…

Let X be a complex Fano manifold of dimension n. Let s(X) be the sum of l(R)-1 for all the extremal rays of X, the edges of the cone NE(X) of curves of X, where l(R) denotes the minimum of (-K_X \cdot C) for all rational curves C whose…

A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. A graph, together with a 1-planar drawing is called 1-plane. Brandenburg et al. showed that there are maximal 1-planar graphs with only…

For a graph class $\mathcal{F}$, let $ex_{\mathcal{F}}(n)$ denote the maximum number of edges in a graph in $\mathcal{F}$ on $n$ vertices. We show that for every proper minor-closed graph class $\mathcal{F}$ the function…

Let $a,b,n$ be three positive integers such that $a\equiv b\pmod 2$ and $n\geq b(a+b)(a+b+2)/(2a)$. Let $G$ be a graph of order $n$ with minimum degree at least $a+b/a-1$. We show that $G$ has an $(a,b)$-parity factor, if…

We study how many copies of a graph $F$ that another graph $G$ with a given number of cliques is guaranteed to have. For example, one of our main results states that for all $t\ge 2$, if $G$ is an $n$ vertex graph with $kn^{3/2}$ triangles…

We prove that one can perfectly pack degenerate graphs into complete or dense $n$-vertex quasirandom graphs, provided that all the degenerate graphs have maximum degree $o(\frac{n}{\log n})$, and in addition $\Omega(n)$ of them have at most…

A graph on $n \ge 3$ vertices drawn in the plane such that each edge is crossed at most four times has at most $6(n-2)$ edges -- this result proven by Ackerman is outstanding in the literature of beyond-planar graphs with regard to its…

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…

Given a graph H, we call a graph $\textit{H-free}$ if it does not contain H as a subgraph. The planar Tur\'an number of a graph H, denoted by $ex_{\mathcal{P}}(n, H)$, is the maximum number of edges in a planar H-free graph on n vertices. A…

Recently, Alon and Frankl (JCTB, 2024) determined the maximum number of edges in $K_{\ell+1}$-free $n$-vertex graphs with bounded matching number. For integers $\ell\ge r \ge 2$, the family $\mathcal{K}_{\ell+1}^{r}$ consists of all…

We combine here Tao's slice-rank bounding method and Gr\"obner basis techniques and apply here to the Erd\H{o}s-Rado Sunflower Conjecture. Let $\frac{3k}{2}\leq n\leq 3k$ be integers. We prove that if $\mbox{$\cal F$}$ be a $k$-uniform…