arXiv subjects

Zixiang Xu

Publications and source records attributed to Zixiang Xu.

At least 73 records · Page 4Linked to original sources

arXiv2024

GenUDC: High Quality 3D Mesh Generation with Unsigned Dual Contouring Representation

Generating high-quality meshes with complex structures and realistic surfaces is the primary goal of 3D generative models. Existing methods typically employ sequence data or deformable tetrahedral grids for mesh generation. However, sequence-based methods have difficulty producing complex structures with many faces due to memory limits. The deformable tetrahedral grid-based method MeshDiffusion fails to recover realistic surfaces due to the inherent ambiguity in deformable grids. We propose the GenUDC framework to address these challenges by leveraging the Unsigned Dual Contouring (UDC) as the mesh representation. UDC discretizes a mesh in a regular grid and divides it into the face and vertex parts, recovering both complex structures and fine details. As a result, the one-to-one mapping between UDC and mesh resolves the ambiguity problem. In addition, GenUDC adopts a two-stage, coarse-to-fine generative process for 3D mesh generation. It first generates the face part as a rough shape and then the vertex part to craft a detailed shape. Extensive evaluations demonstrate the superiority of UDC as a mesh representation and the favorable performance of GenUDC in mesh generation. The code and trained models are available at https://github.com/TrepangCat/GenUDC.

cs.CV↗

arXiv2024

Clique density vs blowups

A well-known theorem of Nikiforov asserts that any graph with a positive $K_{r}$-density contains a logarithmic blowup of $K_r$. In this paper, we explore variants of Nikiforov's result in the following form. Given $r,t\in\mathbb{N}$, when a positive $K_{r}$-density implies the existence of a significantly larger (with almost linear size) blowup of $K_t$? Our results include: For an $n$-vertex ordered graph $G$ with no induced monotone path $P_{6}$, if its complement $\overline{G}$ has positive triangle density, then $\overline{G}$ contains a biclique of size $Ω(\frac{n}{\log{n}})$. This strengthens a recent result of Pach and Tomon. For general $k$, let $g(k)$ be the minimum $r\in \mathbb{N}$ such that for any $n$-vertex ordered graph $G$ with no induced monotone $P_{2k}$, if $\overline{G}$ has positive $K_r$-density, then $\overline{G}$ contains a biclique of size $Ω(\frac{n}{\log{n}})$. Using concentration of measure and the isodiametric inequality on high dimensional spheres, we provide constructions showing that, surprisingly, $g(k)$ grows quadratically. On the other hand, we relate the problem of upper bounding $g(k)$ to a certain Ramsey problem and determine $g(k)$ up to a factor of 2. Any incomparability graph with positive $K_{r}$-density contains a blowup of $K_r$ of size $Ω(\frac{n}{\log{n}}).$ This confirms a conjecture of Tomon in a stronger form. In doing so, we obtain a strong regularity type lemma for incomparability graphs with no large blowups of a clique, which is of independent interest. We also prove that any $r$-comparability graph with positive $K_{(2h-2)^{r}+1}$-density contains a blowup of $K_h$ of size $Ω(n)$, where the constant $(2h-2)^{r}+1$ is optimal. The $\frac{n}{\log n}$ size of the blowups in all our results are optimal up to a constant factor.

math.CO↗

arXiv2024

Beyond chromatic threshold via the $(p,q)$-theorem, and a sharp blow-up phenomenon

We establish a novel connection between the well-known chromatic threshold problem in extremal combinatorics and the celebrated $(p,q)$-theorem in discrete geometry. In particular, for a graph $G$ with bounded clique number and a natural density condition, we prove a $(p,q)$-theorem for an abstract convexity space associated with $G$. Our result strengthens those of Thomassen and Nikiforov on the chromatic threshold of cliques. Our $(p,q)$-theorem can also be viewed as a $χ$-boundedness result for (what we call) ultra maximal $K_r$-free graphs. We further show that the graphs under study are blow-ups of constant size graphs, improving a result of Oberkampf and Schacht on homomorphism threshold of cliques. Our result unravels the cause underpinning such a blow-up phenomenon, differentiating the chromatic and homomorphism threshold problems for cliques. It implies that for the homomorphism threshold problem, rather than the minimum degree condition usually considered in the literature, the decisive factor is a clique density condition on co-neighborhoods of vertices. More precisely, we show that if an $n$-vertex $K_{r}$-free graph $G$ satisfies that the common neighborhood of every pair of non-adjacent vertices induces a subgraph with $K_{r-2}$-density at least $\varepsilon>0$, then $G$ must be a blow-up of some $K_r$-free graph $F$ on at most $2^{O(\frac{r}{\varepsilon}\log\frac{1}{\varepsilon})}$ vertices. Furthermore, this single exponential bound is optimal. We construct examples with no $K_r$-free homomorphic image of size smaller than $2^{Ω_r(\frac{1}{\varepsilon})}$.

math.CO↗

arXiv2024

A polynomial resultant approach to algebraic constructions of extremal graphs

The Turán problem asks for the largest number of edges ex$(n,H)$ in an $n$-vertex graph not containing a fixed forbidden subgraph $H$, which is one of the most important problems in extremal graph theory. However the order of magnitude of ex$(n,H)$ for bipartite graphs is known only in a handful of cases. In particular, giving explicit constructions of extremal graphs is very challenging in this field. In this paper, we develop a polynomail resultant approach to algebraic construction of explicit extremal graphs, which can efficiently decide whether a specified structure exists. A key insight in our approach is the multipolynomial resultant, which is a fundamental tool of computational algebraic geometry. Our main results include the matched lowers bounds for Turán number of $1$-subdivision of $K_{3,t_{1}}$ and linear Turán number of Berge theta hyerpgraph $Θ_{3,t_{2}}^{B}$ with $t_{1}=25$ and $t_{2}=217$. Moreover, the constant $t_{1}$ improves the random algebraic construction of Bukh and Conlon~[Rational exponents in extremal graph theory, J. Eur. Math. Soc. 20 (2018), 1747-1757] and makes progress on the known estimation for the smallest value of $t_{1}$ concerning a problem posed by Conlon, Janzer and Lee ~[More on the extremal number of subdivisions, Combinatorica, to appear], while the constant $t_{2}$ improves a result of He and Tait~[Hypergraphs with few berge paths of fixed length between vertices, SIAM J. Discrete Math., 33(3), 1472-1481].

math.CO↗

arXiv2024

Bollobás-Erdős-Tuza conjecture for graphs with no induced $K_{s,t}$

A widely open conjecture proposed by Bollobás, Erdős, and Tuza in the early 1990s states that for any $n$-vertex graph $G$, if the independence number $α(G) = Ω(n)$, then there is a subset $T \subseteq V(G)$ with $|T| = o(n)$ such that $T$ intersects all maximum independent sets of $G$. In this paper, we prove that this conjecture holds for graphs that do not contain an induced $K_{s,t}$ for fixed $t \ge s$. Our proof leverages the probabilistic method at an appropriate juncture.

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arXiv2024

Piercing independent sets in graphs without large induced matching

Given a graph $G$, denote by $h(G)$ the smallest size of a subset of $V(G)$ which intersects every maximum independent set of $G$. We prove that any graph $G$ without induced matching of size $t$ satisfies $h(G)\le ω(G)^{3t-3+o(1)}$. This resolves a conjecture of Hajebi, Li and Spirkl (Hitting all maximum stable sets in $P_{5}$-free graphs, JCTB 2024).

math.CO↗

arXiv2024

Exact values and improved bounds on $k$-neighborly families of boxes

A finite family $\mathcal{F}$ of $d$-dimensional convex polytopes is called $k$-neighborly if $d-k\le\textup{dim}(C\cap C')\le d-1$ for any two distinct members $C,C'\in\mathcal{F}$. In 1997, Alon initiated the study of the general function $n(k,d)$, which is defined to be the maximum size of $k$-neighborly families of standard boxes in $\mathbb{R}^{d}$. Based on a weighted count of vectors in $\{0,1\}^{d}$, we improve a recent upper bound on $n(k,d)$ by Alon, Grytczuk, Kisielewicz, and Przesławski for any positive integers $d$ and $k$ with $d\ge k+2$. In particular, when $d$ is sufficiently large and $k\ge 0.123d$, our upper bound on $n(k,d)$ improves the bound $\sum_{i=1}^{k}2^{i-1}\binom{d}{i}+1$ shown by Huang and Sudakov exponentially. Furthermore, we determine that $n(2,4)=9$, $n(3,5)=18$, $n(3,6)=27$, $n(4,6)=37$, $n(5,7)=74$, and $n(6,8)=150$. The stability result of Kleitman's isodiametric inequality plays an important role in the proofs.

math.CO↗

arXiv2023

Extremal number of graphs from geometric shapes

We study the Turán problem for highly symmetric bipartite graphs arising from geometric shapes and periodic tilings commonly found in nature. 1. The prism $C_{2\ell}^{\square}:=C_{2\ell}\square K_{2}$ is the graph consisting of two vertex disjoint $2\ell$-cycles and a matching pairing the corresponding vertices of these two cycles. We show that for every $\ell\ge 4$, ex$(n,C_{2\ell}^{\square})=Θ(n^{3/2})$. This resolves a conjecture of He, Li and Feng. 2. The hexagonal tiling in honeycomb is one of the most natural structures in the real world. We show that the extremal number of honeycomb graphs has the same order of magnitude as their basic building unit 6-cycles. 3. We also consider bipartite graphs from quadrangulations of the cylinder and the torus. We prove near optimal bounds for both configurations. In particular, our method gives a very short proof of a tight upper bound for the extremal number of the 2-dimensional grid, improving a recent result of Bradač, Janzer, Sudakov and Tomon. Our proofs mix several ideas, including shifting embedding schemes, weighted homomorphism and subgraph counts and asymmetric dependent random choice.

math.CO↗

arXiv2023

A new variant of the Erdős-Gyárfás problem on $K_{5}$

Motivated by an extremal problem on graph-codes that links coding theory and graph theory, Alon recently proposed a question aiming to find the smallest number $t$ such that there is an edge coloring of $K_{n}$ by $t$ colors with no copy of given graph $H$ in which every color appears an even number of times. When $H=K_{4}$, the question of whether $n^{o(1)}$ colors are enough, was initially emphasized by Alon. Through modifications to the coloring functions originally designed by Mubayi, and Conlon, Fox, Lee and Sudakov, the question of $K_{4}$ has already been addressed. Expanding on this line of inquiry, we further study this new variant of the generalized Ramsey problem and provide a conclusively affirmative answer to Alon's question concerning $K_{5}$.

math.CO↗

arXiv2023

Stability through non-shadows

We study families $\mathcal{F}\subseteq 2^{[n]}$ with restricted intersections and prove a conjecture of Snevily in a stronger form for large $n$. We also obtain stability results for Kleitman's isodiametric inequality and families with bounded set-wise differences. Our proofs introduce a new twist to the classical linear algebra method, harnessing the non-shadows of $\mathcal{F}$, which may be of independent interest.

math.CO↗

arXiv2023

Euclidean Gallai-Ramsey for various configurations

The Euclidean Gallai-Ramsey problem, which investigates the existence of monochromatic or rainbow configurations in a colored $n$-dimensional Euclidean space $\mathbb{E}^{n}$, was introduced and studied recently. We further explore this problem for various configurations including triangles, squares, lines, and the structures with specific properties, such as rectangular and spherical configurations. Several of our new results provide refinements to the results presented in a recent work by Mao, Ozeki and Wang. One intriguing phenomenon evident on the Gallai-Ramsey results proven in this paper is that the dimensions of spaces are often independent of the number of colors. Our proofs primarily adopt a geometric perspective.

math.CO↗

arXiv2023

CTCNet: A CNN-Transformer Cooperation Network for Face Image Super-Resolution

Recently, deep convolution neural networks (CNNs) steered face super-resolution methods have achieved great progress in restoring degraded facial details by jointly training with facial priors. However, these methods have some obvious limitations. On the one hand, multi-task joint learning requires additional marking on the dataset, and the introduced prior network will significantly increase the computational cost of the model. On the other hand, the limited receptive field of CNN will reduce the fidelity and naturalness of the reconstructed facial images, resulting in suboptimal reconstructed images. In this work, we propose an efficient CNN-Transformer Cooperation Network (CTCNet) for face super-resolution tasks, which uses the multi-scale connected encoder-decoder architecture as the backbone. Specifically, we first devise a novel Local-Global Feature Cooperation Module (LGCM), which is composed of a Facial Structure Attention Unit (FSAU) and a Transformer block, to promote the consistency of local facial detail and global facial structure restoration simultaneously. Then, we design an efficient Feature Refinement Module (FRM) to enhance the encoded features. Finally, to further improve the restoration of fine facial details, we present a Multi-scale Feature Fusion Unit (MFFU) to adaptively fuse the features from different stages in the encoder procedure. Extensive evaluations on various datasets have assessed that the proposed CTCNet can outperform other state-of-the-art methods significantly. Source code will be available at https://github.com/IVIPLab/CTCNet.

cs.CV↗

arXiv2023

Local rainbow colorings for various graphs

Motivated by a problem in theoretical computer science suggested by Wigderson, Alon and Ben-Eliezer studied the following extremal problem systematically one decade ago. Given a graph $H$, let $C(n,H)$ be the minimum number $k$ such that the following holds. There are $n$ colorings of $E(K_{n})$ with $k$ colors, each associated with one of the vertices of $K_{n}$, such that for every copy $T$ of $H$ in $K_{n}$, at least one of the colorings that are associated with $V(T)$ assigns distinct colors to all the edges of $E(T)$. In this paper, we obtain several new results in this problem including: \begin{itemize} \item For paths of short length, we show that $C(n,P_{4})=Ω(n^{1/5})$ and $C(n,P_{t})=Ω(n^{1/3})$ with $t\in\{5,6\}$, which significantly improve the previously known lower bounds $(\log{n})^{Ω(1)}$. \item We make progress on the problem of Alon and Ben-Eliezer about complete graphs, more precisely, we show that $C(n,K_{r})=Ω(n^{2/3})$ when $r\geqslant 8$. This provides the first instance of graph for which the lower bound goes beyond the natural barrier $Ω(n^{1/2})$. Moreover, we prove that $C(n,K_{s,t})=Ω(n^{2/3})$ for $t\geqslant s\geqslant 7$. \item When $H$ is a star with at least $4$ leaves, a matching of size at least $4$, or a path of length at least $7$, we give the new lower bound for $C(n,H)$. We also show that for any graph $H$ with at least $6$ edges, $C(n,H)$ is polynomial in $n$. All of these improve the corresponding results obtained by Alon and Ben-Eliezer.

math.CO↗

arXiv2022

A generic framework for coded caching and distributed computation schemes

Several network communication problems are highly related such as coded caching and distributed computation. The centralized coded caching focuses on reducing the network burden in peak times in a wireless network system and the coded distributed computation studies the tradeoff between computation and communication in distributed system. In this paper, motivated by the study of the only rainbow $3$-term arithmetic progressions set, we propose a unified framework for constructing coded caching schemes. This framework builds bridges between coded caching schemes and lots of combinatorial objects due to the freedom of the choices of families and operations. We prove that any scheme based on a placement delivery array (PDA) can be represented by a rainbow scheme under this framework and lots of other known schemes can also be included in this framework. Moreover, we also present a new coded caching scheme with linear subpacketization and near constant rate using the only rainbow $3$-term arithmetic progressions set. Next, we modify the framework to be applicable to the distributed computing problem. We present a new transmission scheme in the shuffle phase and show that in certain cases it could have a lower communication load than the schemes based on PDAs or resolvable designs with the same number of files.

cs.IT↗

arXiv2022

An Explore of Virtual Reality for Awareness of the Climate Change Crisis: A Simulation of Sea Level Rise

Virtual Reality (VR) technology has been shown to achieve remarkable results in multiple fields. Due to the nature of the immersive medium of Virtual Reality it logically follows that it can be used as a high-quality educational tool as it offers potentially a higher bandwidth than other mediums such as text, pictures and videos. This short paper illustrates the development of a climate change educational awareness application for virtual reality to simulate virtual scenes of local scenery and sea level rising until 2100 using prediction data. The paper also reports on the current in progress work of porting the system to Augmented Reality (AR) and future work to evaluate the system.

cs.MM↗

arXiv2021

A note on multicolor Ramsey number of small odd cycles versus a large clique

Let $R_k(H;K_m)$ be the smallest number $N$ such that every coloring of the edges of $K_{N}$ with $k+1$ colors has either a monochromatic $H$ in color $i$ for some $1\leqslant i\leqslant k$, or a monochromatic $K_{m}$ in color $k+1$. In this short note, we study the lower bound for $R_k(H;K_m)$ when $H$ is $C_5$ or $C_7$, respectively. We show that \begin{equation*} R_{k}(C_5;K_m)=Ω(m^{\frac{3k}{8}+1}/(\log{m})^{\frac{3k}{8}+1}), \end{equation*} and \begin{equation*} R_{k}(C_7;K_m)=Ω(m^{\frac{2k}{9}+1}/(\log{m})^{\frac{2k}{9}+1}), \end{equation*} for fixed positive integer $k$ and $m\rightarrow\infty$. These slightly improve the previously known lower bound $R_{k}(C_{2\ell+1};K_m)=Ω(m^{\frac{k}{2\ell-1}+1}/(\log m)^{k+\frac{2k}{2\ell-1}})$ obtained by Alon and Rödl. The proof is based on random block constructions and random blowups argument.

math.CO↗

arXiv2021

Some extremal results on hypergraph Turán problems

For two $r$-graphs $\mathcal{T}$ and $\mathcal{H}$, let $\text{ex}_{r}(n,\mathcal{T},\mathcal{H})$ be the maximum number of copies of $\mathcal{T}$ in an $n$-vertex $\mathcal{H}$-free $r$-graph. The determination of Turán number $\text{ex}_{r}(n,\mathcal{T},\mathcal{H})$ has become the fundamental core problem in extremal graph theory ever since the pioneering work Turán's Theorem was published in $1941$. Although we have some rich results for the simple graph case, only sporadic results have been known for the hypergraph Turán problems. In this paper, we mainly focus on the function $\text{ex}_{r}(n,\mathcal{T},\mathcal{H})$ when $\mathcal{H}$ is one of two different hypergraph extensions of the complete bipartite graph $K_{s,t}$. The first extension is the complete bipartite $r$-graph $K_{s,t}^{(r)}$, which was introduced by Mubayi and Verstraëte~[J. Combin. Theory Ser. A, 106: 237--253, 2004]. Using the powerful random algebraic method, we show that if $s$ is sufficiently larger than $t$, then \[\text{ex}_{r}(n,\mathcal{T},K_{s,t}^{(r)})=Ω(n^{v-\frac{e}{t}}),\] where $\mathcal{T}$ is an $r$-graph with $v$ vertices and $e$ edges. In particular, when $\mathcal{T}$ is an edge or some specified complete bipartite $r$-graph, we can determine their asymptotics. The second important extension is the complete $r$-partite $r$-graph $K_{s_{1},s_{2},\ldots,s_{r}}^{(r)}$, which has been widely studied. When $r=3$, we provide an explicit construction giving \[\text{ex}_{3}(n,K_{2,2,7}^{(3)})\geqslant\frac{1}{27}n^{\frac{19}{7}}+o(n^{\frac{19}{7}}).\] Our construction is based on the Norm graph, and improves the lower bound $Ω(n^{\frac{73}{27}})$ obtained by probabilistic method.

math.CO↗

arXiv2020

On color isomorphic subdivisions

Given a graph $H$ and an integer $k\geqslant 2$, let $f_{k}(n,H)$ be the smallest number of colors $C$ such that there exists a proper edge-coloring of the complete graph $K_{n}$ with $C$ colors containing no $k$ vertex-disjoint color isomorphic copies of $H$. In this paper, we prove that $f_{2}(n,H_{t})=Ω(n^{1+\frac{1}{2t-3}})$ where $H_{t}$ is the $1$-subdivision of the complete graph $K_{t}$. This answers a question of Conlon and Tyomkyn (arXiv: 2002.00921).

math.CO↗