arXiv subjects

Haorui Liu

Publications and source records attributed to Haorui Liu.

3 recordsLinked to original sources

arXiv2026

A Quadratic Vertex Threshold for Isolated Cliques in the Minimum Degree Kruskal-Katona Problem for 3-Uniform Hypergraphs

Given a set $X$ and an integer $t$, let $\mathcal{F}$ be a family of $k$-subsets of $X$. The Kruskal-Katona theorem states that if $|\mathcal{F}|\geq \binom{t}{k}$, then $|\partial_{k-1}\mathcal{F}|\geq\binom{t}{k-1}$. The minimum degree version of this problem asks: if $\delta(\mathcal{F})\geq \binom{t}{k-1}$, how small can $|\partial_{k-1}\mathcal{F}|$ be? In this article, for the case $k=3$, we prove that, for every sufficiently large integer $t$, every extremal hypergraph for this problem contains an isolated copy of $K_{t+1}^3$ whenever $|X| \geq ct^2 + o(t^2)$, with the constant $c = 1 + \sqrt{928/33}$. Our proof uses a graph transformation that regularizes the neighborhood structure of extremal graphs, reducing the problem to a counting argument on the neighbors of a disjoint clique family. This gives a quadratic-order threshold for the every-extremal version of the problem, compared with the cubic-order threshold of F\"{u}redi and Zhao [SIAM J.\ Discrete Math.\ 36(4), 2022].

math.CO↗

arXiv2026

Shadows of Uniform Hypergraphs under a Minimum Degree Condition

Given a set $X$ and an integer $t$, let $\mathcal{F}$ be a family of $k$-subsets of $X$. The Kruskal--Katona theorem implies that if $|\mathcal{F}|\geq \binom{t}{k}$, then $|\partial_\ell\mathcal{F}|\geq\binom{t}{\ell}$. The minimum degree version of this problem asks: if $\delta(\mathcal{F})\geq \binom{t}{k-1}$, how small can $|\partial_\ell\mathcal{F}|$ be? We call a hypergraph \textit{extremal} if it achieves the minimum value of $|\partial_\ell \mathcal{F}|$ subject to the degree condition $\delta(\mathcal{F}) \geq \binom{t}{k-1}$. F\"uredi and Zhao [SIAM J. Discrete Math. 36(4), 2022] proved that for $k=3$, $\ell=2$ and $t\ge 2$, every extremal hypergraph contains an isolated copy of $K_{t+1}^3$ when $|X| > \frac{1}{4}(t+1)^2(t+2)$. In this article, we study the general case $k > \ell \geq 2$. By developing a hypergraph transformation that combines shifting operations with antilexicographic compression, we prove that, for every integer $t\ge k-1$, there exists an extremal hypergraph containing an isolated copy of $K^{k}_{t+1}$ whenever $|X| > \frac{1}{4}(t+1)^2\binom{t-1}{\ell-2} + 3t+1$. In the case when $k=3$ and $\ell=2$, this gives the threshold $\frac14(t+1)^2+3t+1$, which is smaller than $\frac14(t+1)^2(t+2)$ for every $t\ge3$; for $t=2$, the two thresholds give the same integer condition on $|X|$.

math.CO↗

arXiv2025

Vertex degree sums for rainbow matchings in 3-uniform hypergraphs

Let $n \in 3\mathbb{Z}$ be sufficiently large. Zhang, Zhao and Lu proved that if $H$ is a 3-uniform hypergraph with $n$ vertices and no isolated vertices, and if $deg(u)+deg(v) > \frac{2}{3}n^2 - \frac{8}{3}n + 2$ for any two vertices $u$ and $v$ that are contained in some edge of $H$, then $ H $ admits a perfect matching. In this paper, we prove that the rainbow version of Zhang, Zhao and Lu's result is asymptotically true. More specifically, let $\delta > 0$ and $ F_1, F_2, \dots, F_{n/3} $ be 3-uniform hypergraphs on a common set of $n$ vertices. For each $ i \in [n/3] $, suppose that $F_i$ has no isolated vertices and $deg_{F_i}(u)+deg_{F_i}(v) > \left( \frac{2}{3} + \delta \right)n^2$ holds for any two vertices $u$ and $v$ that are contained in some edge of $F_i$. Then $ \{ F_1, F_2, \dots, F_{n/3} \} $ admits a rainbow matching. Note that this result is asymptotically tight.

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