arXiv subjects

Donglei Yang

Publications and source records attributed to Donglei Yang.

At least 19 recordsLinked to original sources

arXiv2026

Transversal tilings in k-partite graphs without large holes

We show that for any constant $\mu>0$ and $k\ge 3$, there exists $\alpha>0$ such that the following holds for sufficiently large $n \in \mathbb{N}$. If $G=(V_{1},\ldots,V_{k},E)$ is a spanning subgraph of the $n$-blow-up of $K_{k}$ with ${\delta^*}(G)\geq (\frac{1}{2}+\mu) n$ and $\alpha^*_{k-1}(G)<\alpha n$, then $G$ has a transversal $K_{k}$-factor. Moreover, the bound $\frac{1}{2}$ is asymptotically tight for the case $k=3$. In addition, we show that if $k\ge 4$, $G=(V_{1},\ldots,V_{k},E)$ is a spanning subgraph of the $n$-blow-up of $C_{k}$ with ${\delta^*}(G)\ge (\frac{2}{k}+\mu) n$, and $\alpha^*_{2}(G)<\alpha n$, then $G$ has a transversal $C_{k}$-factor. This extends a recent result of Han, Hu, Ping, Wang, Wang and Yang.

math.CO↗

arXiv2025

Clique factors in random samplings of regular graphs

We show that for any integer $r\ge 2$, there exists a constant $c>0$ such that for every sufficiently large integer $n$, every $((r-1)n+1)$-regular graph $G$ on $rn$ vertices has at least $c2^{rn}$ subsets $S\subseteq V(G)$ such that $G[S]$ contains a $K_r$-factor. This confirms a conjecture of Dragani\'c, Keevash and M\"uyesser for large $n$ [Cyclic subsets in regular Dirac graphs. Int. Math. Res. Not., 2025(14): 1-16, 2025].

math.CO↗

arXiv2025

Dense minors and bipartite independence numbers

A graph $G$ is $m$-joined if there is an edge between every two disjoint $m$-sets of vertices. In this paper, we prove that for any $\varepsilon>0$ and sufficiently large $m, n\in \mathbb{N}$ with $m \le n^{1-\varepsilon}$, every $n$-vertex $m$-joined graph $G$ contains a minor with density $\Omega\!\left(\tfrac{n}{\sqrt{m}}\right)$, which is best possible up to a constant factor. When $m \ge n^{1-\varepsilon}$, we further show that $G$ contains a clique minor of order $\Omega\!\left(\tfrac{n}{\sqrt{m\log m}}\right)$.

math.CO↗

arXiv2025

Clique-factors in graphs with low $K_{\ell}$-independence number

Given $r\in \mathbb{N}$ with $r\geq 4$, we show that there exists $n_0\in \mathbb{N}$ such that for every $n\geq n_0$, every $n$-vertex graph $G$ with $\delta(G)\geq (\frac{1}{2}+o(1))n$ and $\alpha_{r-2}(G)=o(n)$ contains a $K_{r}$-factor. This resolves the first open case of a question proposed by Nenadov and Pehova, and reiterated by Knierm and Su. We further introduce two lower bound constructions that, along with some known results, fully resolve a question presented by Balogh, Molla, and Sharifzadeh.

math.CO↗

arXiv2025

Perfect tilings with the generalised triangle in $k$-graphs

Denote by $T_k$ the generalised triangle, a $k$-uniform hypergraph on vertex set $\{1,2,\dots,2k-1\}$ with three edges $\{1,\dots,k-1,k\}$,$\{1,\dots,k-1,k+1\}$ and $\{k,k+1,\dots,2k-1\}$. Recently, Bowtell, Kathapurkar, Morrison and Mycroft [arXiv: 2505.05606] established the exact minimum codegree threshold for perfect $T_3$-tilings in $3$-graphs. In this paper, we extend their result to all $k \geq 3$, determining the optimal minimum codegree threshold for perfect $T_k$-tilings in $k$-graphs. Our proof uses the lattice-based absorption method, as is usual, but develops a unified and effective approach to build transferrals for all uniformities, which is of independent interest. Additionally, we establish an asymptotically tight minimum codegree threshold for a rainbow variant of the problem.

math.CO↗

arXiv2025

Transversal packings in families of percolated hypergraphs

Let $F$ be a strictly $1$-balanced $k$-graph on $s$ vertices with $t$ edges and $\delta_{F,d}^T$ be the infimum of $\delta>0$ such that for every $\alpha>0$ and sufficiently large $n\in \mathbb{N}$, every $k$-graph system $\mathbf H=\{H_{1}, H_{2}, \dots ,H_{tn}\}$ on the same $sn$ vertices with $\delta_d(H_i)\ge (\delta+\alpha)\binom{sn-d}{k-d}$, $i\in [tn]$ contains a transversal $F$-factor, that is, an $F$-factor consisting of exactly one edge from each $H_i$. In this paper we prove the following result. Let $\mathbf{H} =\{H_{1}, H_{2}, \dots ,H_{tn}\}$ be a $k$-graph system where each $H_{i}$ is an $sn$-vertex $k$-graph with $\delta_d(H_i)\ge (\delta_{F,d}^T+\alpha)\binom{sn-d}{k-d}$. Then with high probability $\mathbf{H}(p) :=\{H_{1}(p), H_{2}(p), \dots ,H_{tn}(p)\}$ contains a transversal $F$-factor, where $H_i(p)$ is a random subhypergraph of $H_i$ and $p=\Omega(n^{-1/d_1(F)-1}(\log n)^{1/t})$. This extends a recent result by Kelly, M\"{u}yesser and Pokrovskiy, and independently by Joos, Lang and Sanhueza-Matamala. Moreover, the assumption on $p$ is best possible up to a constant. Along the way, we also obtain a spread version of a result of Pikhurko on perfect matchings in $k$-partite $k$-graphs.

math.CO↗

arXiv2025

Modulating lipid membrane morphology by dynamic DNA origami networks

Membrane morphology and its dynamic adaptation regulate many cellular functions, which are often mediated by membrane proteins. Advances in DNA nanotechnology have enabled the realization of various protein-inspired structures and functions with precise control at the nanometer level, suggesting a viable tool to artificially engineer the membrane morphology. In this work, we demonstrate a DNA origami cross (DOC) structure that can be anchored onto giant unilamellar vesicles (GUVs) and subsequently polymerized into micron-scale reconfigurable one-dimensional (1D) chains or two-dimensional (2D) lattices. Such DNA origami-based networks can be switched between left-handed (LH) and right-handed (RH) conformations by DNA fuels and exhibit potent efficacy in remodeling the membrane curvatures of GUVs. This work sheds light on designing hierarchically-assembled dynamic DNA systems for the programmable modulation of synthetic cells for useful applications.

physics.bio-ph↗

arXiv2025

Packing tetrahedrons in edge-weighted graphs

We prove that for all $\mu>0, t\in (0,1)$ and sufficiently large $n\in 4\mathbb{N}$, if $G$ is an edge-weighted complete graph on $n$ vertices with a weight function $w: E(G)\rightarrow [0,1]$ and the minimum weighted degree $\delta^w(G)\geq (\tfrac{1+3t}{4}+\mu)n$, then $G$ contains a $K_4$-factor where each copy of $K_4$ has total weight more than $6t$. This confirms a conjecture of Balogh--Kemkes--Lee--Young for the tetrahedron case.

math.CO↗

arXiv2025

Extremal density for subdivisions with length or sparsity constraints

Given a graph $H$, a balanced subdivision of $H$ is obtained by replacing all edges of $H$ with internally disjoint paths of the same length. In this paper, we prove that for any graph $H$, a linear-in-$e(H)$ bound on average degree guarantees a balanced $H$-subdivision. This strengthens an old result of Bollobás and Thomason, and resolves a question of Gil-Fernández, Hyde, Liu, Pikhurko and Wu. We observe that this linear bound on average degree is best possible whenever $H$ is logarithmically dense. We further show that this logarithmic density is the critical threshold: for many graphs $H$ below this density, its subdivisions are forcible by a sublinear-in-$e(H)$ bound on average degree. We provide such examples by proving that the subdivisions of any almost bipartite graph $H$ with sublogarithmic density are forcible by a sublinear-in-$e(H)$ bound on average degree, provided that $H$ satisfies some additional separability condition.

math.CO↗

arXiv2024

Ramsey--Dirac theory for bounded degree hypertrees

Ramsey--Turán theory considers Turán type questions in Ramsey-context, asking for the existence of a small subgraph in a graph $G$ where the complement $\overline{G}$ lacks an appropriate subgraph $F$, such as a clique of linear size. Similarly, one can consider Dirac-type questions in Ramsey context, asking for the existence of a spanning subgraph $H$ in a graph $G$ where the complement $\overline{G}$ lacks an appropriate subgraph $F$, which we call a Ramsey--Dirac theory question. When $H$ is a connected spanning subgraph, the disjoint union $K_{n/2}\cup K_{n/2}$ of two large cliques shows that it is natural to consider complete bipartite graphs $F$. Indeed, Han, Hu, Ping, Wang, Wang and Yang in 2024 proved that if $G$ is an $n$-vertex graph with $δ(G)=Ω(n)$ where the complement $\overline{G}$ does not contain any complete bipartite graph $K_{m,m}$ with $m=Ω(n)$, then $G$ contains every $n$-vertex bounded degree tree $T$ as a subgraph. Extending this result to the Ramsey--Dirac theory for hypertrees, we prove that if $G$ is an $n$-vertex $r$-uniform hypergraph with $δ(G)=Ω(n^{r-1})$ where the complement $\overline{G}$ does not contain any complete $r$-partite hypergraph $K_{m,m,\dots, m}$ with $m=Ω(n)$, then $G$ contains every $n$-vertex bounded degree hypertree $T$ as a subgraph. We also prove the existence of matchings and loose Hamilton cycles in the same setting, which extends the result of Mcdiarmid and Yolov into hypergraphs. This result generalizes the universality result on randomly perturbed graphs by Böttcher, Han, Kohayakawa, Montgomery, Parczyk and Person in 2019 into hypergraphs and also strengthen the results on quasirandom hypergraphs by Lenz, Mubayi and Mycroft in 2016 and Lenz and Mubayi in 2016 into hypergraphs satisfying a much weaker pseudorandomness condition.

math.CO↗

arXiv2024

Topological cliques in sparse expanders

In the paper, we focus on embedding clique immersions and subdivisions within sparse expanders, and we derive the following main results: (1) For any $0< η< 1/2$, there exists $K>0$ such that for sufficiently large $n$, every $(n,d,λ)$-graph $G$ contains a $K_{(1-5η)d}$-immersion when $d\geq Kλ$. (2) For any $\varepsilon>0$ and $0<η<1/2$, the following holds for sufficiently large $n$. Every $(n,d,λ)$-graph $G$ with $2048λ/η^2 0$ such that the following holds for sufficiently large $d$. If $G$ is an $n$-vertex graph with average degree $d(G)\geq d$, then $G$ contains a $K_{c d}^{(\ell)}$-immersion for some $\ell\in \mathbb{N}$. In 2018, Dvo{ř}{á}k and Yepremyan asked whether every graph $G$ with $δ(G)\geq t$ contains a $K_t$-immersion. Our first result shows that it is asymptotically true for $(n,d,λ)$-graphs when $λ=o(d)$. In addition, our second result extends a result of Dragani{ć}, Krivelevich and Nenadov on balanced subdivisions. The last result generalises a result of DeVos, Dvo{ř}{á}k, Fox, McDonald, Mohar, Scheide on $1$-immersions of large cliques in dense graphs.

math.CO↗

arXiv2024

Balanced clique subdivisions and cycles lengths in $K_{s, t}$-free graphs

Let $ t\ge s\ge2$ be integers. Confirming a conjecture of Mader, Liu and Montgomery [J. Lond. Math. Soc., 2017] showed that every $K_{s, t}$-free graph with average degree $d$ contains a subdivision of a clique with at least $\Omega(d^{\frac{s}{2(s-1)}})$ vertices. We give an improvement by showing that such a graph contains a balanced subdivision of a clique with the same order, where a balanced subdivision is a subdivision in which each edge is subdivided the same number of times. In 1975, Erd\H{o}s asked whether the sum of the reciprocals of the cycle lengths in a graph with infinite average degree $d$ is necessarily infinite. Recently, Liu and Montgomery [J. Amer. Math. Soc., 2023] confirmed the asymptotically correct lower bound on the reciprocals of the cycle lengths, and provided a lower bound of at least $(\frac{1}{2} -o_d(1)) \log d$. In this paper, we improve this low bound to $\left(\frac{s}{2(s-1)} -o_d(1)\right) \log d$ for $K_{s, t}$-free graphs. Both proofs of our results use the graph sublinear expansion property as well as some novel structural techniques.

math.CO↗

arXiv2024

Embedding clique subdivisions via crux

For a graph $G$ and a constant $α>0$, we denote by $C_α(G)$ the minimum order of a subgraph $H\subseteq G$ with $d(H)\ge αd(G)$. Liu and Montgomery conjectured that every graph $G$ contains $K_{Ω(t)}$ as a subdivision for $t=\min \{d(G), \sqrt{\tfrac{C_α(G)}{\log C_α(G)}}\}$. In the paper, we prove this conjecture.

math.CO↗

arXiv2023

A robust version of the multipartite Hajnal--Szemerédi theorem

In this note we show the following strengthening of a multipartite version of the Hajnal--Szemerédi theorem. For an integer $r \ge 3$ and $γ> 0$, there exists a constant $C$ such that if $p\ge Cn^{-2/r}(\log n)^{1/{r \choose 2}}$ and $G$ is a balanced $r$-partite graph with each vertex class of size $n$ and $δ^\ast(G)\ge (1-1/r+γ)n$, then with high probability the random subgraph $G(p)$ of $G$ contains a $K_r$-factor. We also use it to derive corresponding transversal versions.

math.CO↗

arXiv2023

On powers of Hamilton cycles in Ramsey-Turán Theory

We prove that for $r\in \mathbb{N}$ with $r\geq 2$ and $μ>0$, there exist $α>0$ and $n_{0}$ such that for every $n\geq n_{0}$, every $n$-vertex graph $G$ with $δ(G)\geq \left(1-\frac{1}{r}+μ\right)n$ and $α(G)\leq αn$ contains an $r$-th power of a Hamilton cycle. We also show that the minimum degree condition is asymptotically sharp for $r=2, 3$ and the $r=2$ case was recently conjectured by Staden and Treglown.

math.CO↗

arXiv2023

Transversal Hamilton cycle in hypergraph systems

A $k$-graph system $\textbf{H}=\{H_i\}_{i\in[m]}$ is a family of not necessarily distinct $k$-graphs on the same $n$-vertex set $V$ and a $k$-graph $H$ on $V$ is said to be $\textbf{H}$-transversal provided that there exists an injection $φ: E(H)\rightarrow [m]$ such that $e\in E(H_{φ(e)})$ for all $e\in E(H)$. We show that given $k\geq3, γ>0$, sufficiently large $n$ and an $n$-vertex $k$-graph system $\textbf{H}=\{H_i\}_{i\in[n]}$, if $δ_{k-1}(H_i)\geq(1/2+γ)n$ for each $i\in[n]$, then there exists an $\textbf{H}$-transversal tight Hamilton cycle. This extends the result of Rödl, Ruciński and Szemerédi [Combinatorica, 2008] on single $k$-graphs.

math.CO↗

arXiv2023

Graph tilings in incompatibility systems

An \emph{incompatibility system} $(G,\mathcal{F})$ consists of a graph $G$ and a family $\mathcal{F}=\{F_v\}_{v\in V(G)}$ over $G$ with $F_v\subseteq \{\{e,e'\}\in {E(G)\choose 2}: e\cap e'=\{v\}\}$. We say that two edges $e,e'\in E(G)$ are \emph{incompatible} if $\{e,e'\}\in F_v$ for some $v\in V(G)$, and otherwise \emph{compatible}. A subgraph $H$ of $G$ is \emph{compatible} if every pair of edges in $H$ are compatible. An incompatibility system $(G,\mathcal{F})$ is \emph{$Δ$-bounded} if for any vertex $v$ and any edge $e$ incident with $v$, there are at most $Δ$ members of $F_v$ containing $e$. This notion was partly motivated by a concept of transition system introduced by Kotzig in 1968, and first formulated by Krivelevich, Lee and Sudakov to study the robustness of Hamiltonicity of Dirac graphs. We prove that for any $α>0$ and any graph $H$ with $h$ vertices, there exists a constant $μ>0$ such that for any sufficiently large $n$ with $n\in h\mathbb{N}$, if $G$ is an $n$-vertex graph with $δ(G)\ge(1-\frac{1}{χ^*(H)}+α)n$ and $(G,\mathcal{F})$ is a $μn$-bounded incompatibility system, then there exists a compatible $H$-factor in $G$, where the value $χ^*(H)$ is either the chromatic number $χ(H)$ or the critical chromatic number $χ_{cr}(H)$ and we provide a dichotomy as in the Kühn--Osthus result. Moreover, we give examples $H$ for which there exists an $μn$-bounded incompatibility system $(G, \mathcal{F})$ with $n\in h\mathbb{N}$ and $δ(G)\ge(1-\frac{1}{χ^*(H)}+\fracμ{2})n$ such that $G$ contains no compatible $H$-factor. Unlike in the previous work of Kühn and Osthus on embedding $H$-factors, our proof uses the lattice-based absorption method.

math.CO↗

arXiv2023

Clique-factors in graphs with sublinear $\ell$-independence number

Given a graph $G$ and an integer $\ell\ge 2$, we denote by $α_{\ell}(G)$ the maximum size of a $K_{\ell}$-free subset of vertices in $V(G)$. A recent question of Nenadov and Pehova asks for determining the best possible minimum degree conditions forcing clique-factors in $n$-vertex graphs $G$ with $α_{\ell}(G) = o(n)$, which can be seen as a Ramsey--Turán variant of the celebrated Hajnal--Szemerédi theorem. In this paper we find the asymptotical sharp minimum degree threshold for $K_r$-factors in $n$-vertex graphs $G$ with $α_\ell(G)=n^{1-o(1)}$ for all $r\ge \ell\ge 2$.

math.CO↗