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Zilong Yan

Publications and source records attributed to Zilong Yan.

10 recordsLinked to original sources
arXiv2026

Finite palette endpoints and degree-square Tur\'an problems

We study finite palette extremal problems motivated by uniform Tur\'an densities of $3$-uniform hypergraphs. Given a self-converse tournament $T$ with at least two vertices, we determine the largest possible number of admissible triples in an $m$-color palette that avoids the left and right palettes associated with $T$. The answer is the one-sided degree-square Tur\'an number \[ \operatorname{pal}_T(m) = \operatorname{ex}_2^+(m,T) = \max\left\{ \sum_{v\in V(D)} d_D^+(v)^2: |V(D)|=m,\ D\text{ is }T\text{-free} \right\}. \] Thus this palette problem is reduced to an extremal problem for digraph out-degrees. We then prove a prefix-majorization lemma for convex out-degree moments and apply it to directed cycles. In particular, $\operatorname{ex}_2^+(m,\overrightarrow C_3)=\frac{m(m^2-1)}3$, which gives the sharp $m$-color palette endpoint $\frac13-\frac{1}{3m^2}$ for the directed triangle. Combining this endpoint with the palette characterization of uniform Tur\'an density and the palette separation theorem, we show that for every $m\ge2$ there is a finite $3$-graph $H_m$ such that \[ \frac13-\frac{1}{3m^2}\le \pi_u(H_m)\le \frac13. \] Hence there is a sequence of individual finite $3$-graphs whose uniform Tur\'an densities converge to $1/3$. We also describe the extremal palettes, prove a qualitative edit-distance stability theorem, and compute the Lagrangian of the endpoint palette $\mathcal P_m^\star$. As a consequence, for every $m\ge2$ there is a finite family $\mathcal F_m$ of $3$-graphs with $\pi_u(\mathcal F_m)=\frac13-\frac{1}{3m^2}$, so $1/3$ is an accumulation point of uniform Tur\'an densities of finite forbidden families.

math.CO↗
arXiv2024

Chromatic profiles of odd cycles

Erd\H{o}s and Simonovits asked the following question: For an integer $c\geq 2$ and a family of non-bipartite graphs $\mathcal{F}$, what is the infimum of $\alpha$ such that any $\mathcal{F}$-free $n$-vertex graph with $n$ large enough and minimum degree at least $\alpha n$ has chromatic number at most $c$? Denote the infimum as $\delta_{\chi}(\mathcal{F}, c)$. A fundamental result of Erd\H{o}s, Stone and Simonovits implies that if $3\le r+1=\chi(\mathcal{F})=\min\{\chi (F): F\in \mathcal{F}\}$, then for any $c\le r-1$, $\delta_{\chi}(\mathcal{F}, c)=1-{1 \over r}$. So the remaining challenge is to determine $\delta_{\chi}(\mathcal{F}, c)$ for $c\ge \chi (\mathcal{F})-1$. Most previous known results are under the condition that $c= \chi (\mathcal{F})-1$. When $c\ge \chi (\mathcal{F})$, the only known exact results are $\delta_{\chi}(K_3, 3)$ by H\"aggkvist and Jin, and $\delta_{\chi}(K_3, c)$ for every $c\ge4$ by Brandt and Thomass\'{e}, $\delta_{\chi}(K_r, r)$ and $\delta_{\chi}(K_r, r+1)$ by Goddard and Lyle, and Nikiforov. Combining results of Thomassen and Ma, $\Omega\bigg((c+1)^{-8(k+1)}\bigg)=\delta_{\chi}(C_{2k+1}, c)=O(\frac{k}{c})$ for $c\ge 3$. In this paper, we determine $\delta_{\chi}(C_{2k+1}, c)$ for all $c\ge 2$ and $k\ge 3c+4$. We also obtain the following corollary. If $G$ is a graph on $n$ vertices with $c\ge 3$, $\chi(G)>c$ and $\delta(G)> {n \over 2c+2}$, then $C_{2k+1} \subset G$ for all $k\in [3c+4, {n \over 108(c+1)^c}]$. Methods to obtain all previous known results related to odd cycles cannot be applied to solve for $\delta_{\chi}(C_{2k+1}, c)$ for $c\ge 3$.The innovation of our proof is to give the concept of a `strong $2k$-core'. We think that this concept grasps the essence of the problem and it makes our proof concise and elementary (we do not need to borrow any other tools). How to define a proper `core' might be a key to this type of questions.

math.CO↗
arXiv2024

A strong structural stability of $C_{2k+1}$-free graphs

F\"uredi and Gunderson showed that $ex(n, C_{2k+1})$ is achieved only on $K_{\lfloor\frac{n}{2}\rfloor, \lceil\frac{n}{2}\rceil}$ if $n\ge 4k-2$. It is natural to study how far a $ C_{2k+1}$-free graph is from being bipartite.Let $T^*(r, n)$ be obtained by adding a suspension $K_{r}$ with $1$ suspension point to $K_{\lfloor\frac{n-r+1}{2}\rfloor, \lceil\frac{n-r+1}{2}\rceil}$. We show that for integers $r, k$ with $3\le r\le 2k-4$ and $n\ge 20(r+2)^2k$, if $G$ is a $C_{2k+1}$-free $n$-vertex graph with $e(G)\ge e(T^*(r, n))$, then $G$ is obtained by adding suspensions to a bipartite graph one by one and the total number of vertices in all suspensions minus intersection points is no more than $r-1$. In other words, $G=B\bigcup\limits_{i=1}^p G_i$, where $B$ is a bipartite graph, $G_1$ is a suspension to $B$, $G_j$ is a suspension to $B\bigcup\limits_{i=1}^{j-1} G_i$ for $2\le j\le p$ and $\sum\limits_{i=1}^p \vert V(G_i)-V(G_i)\cap V(B\bigcup\limits_{i=1}^{j-1} G_i) \vert\le r-1$. Furthermore, $\sum\limits_{i=1}^p \vert V(G_i)-V(G_i)\cap V(B\bigcup\limits_{i=1}^{j-1} G_i) \vert= r-1$ if and only if $G=T^*(r, n)$. Let $d_2(G)=\min\{|T|: T\subseteq V(G), G-T \ \text{is bipartite}\}$ and $\gamma_2(G)=\min\{|E|: E\subseteq E(G), G-E \ \text{is bipartite}\}$. Our structural stability result implies that $d_2(G)\le r-1$ and $\gamma_2(G)\le {\lceil\frac{r}{2}\rceil \choose 2}+{\lfloor\frac{r}{2}\rfloor \choose 2}$ under the same condition, which is a recent result of Ren-Wang-Wang-Yang [SIAM J. Discrete Math. 38 (2024)]. They proved $d_2(G)\le r-1$ and $\gamma_2(G)\le {\lceil\frac{r}{2}\rceil \choose 2}+{\lfloor\frac{r}{2}\rfloor \choose 2}$ separately. We introduce a new concept strong-$2k$-core which is the key that we can give a stronger structural stability result but a simpler proof.

math.CO↗
arXiv2023

SAI: Solving AI Tasks with Systematic Artificial Intelligence in Communication Network

In the rapid development of artificial intelligence, solving complex AI tasks is a crucial technology in intelligent mobile networks. Despite the good performance of specialized AI models in intelligent mobile networks, they are unable to handle complicated AI tasks. To address this challenge, we propose Systematic Artificial Intelligence (SAI), which is a framework designed to solve AI tasks by leveraging Large Language Models (LLMs) and JSON-format intent-based input to connect self-designed model library and database. Specifically, we first design a multi-input component, which simultaneously integrates Large Language Models (LLMs) and JSON-format intent-based inputs to fulfill the diverse intent requirements of different users. In addition, we introduce a model library module based on model cards which employ model cards to pairwise match between different modules for model composition. Model cards contain the corresponding model's name and the required performance metrics. Then when receiving user network requirements, we execute each subtask for multiple selected model combinations and provide output based on the execution results and LLM feedback. By leveraging the language capabilities of LLMs and the abundant AI models in the model library, SAI can complete numerous complex AI tasks in the communication network, achieving impressive results in network optimization, resource allocation, and other challenging tasks.

cs.AI↗
arXiv2023

The bipartite Ramsey number $br(C_{2n}, C_{2m})$

Given bipartite graphs $H_1$, \dots , $H_k$, the bipartite Ramsey number $br(H_1,\dots, H_k)$ is the minimum integer $N$ such that any $k$-edge-coloring of complete bipartite graph $K_{N, N}$ contains a monochromatic $H_i$ in color $i$ for $1\le i\le k$. There are considerable results on asymptotic values of bipartite Ramsey numbers of cycles. For exact value, Zhang-Sun \cite{Zhangs} determined $br(C_4, C_{2n})$, Zhang-Sun-Wu \cite{Zhangsw} determined $br(C_6, C_{2n})$, and Gholami-Rowshan \cite{GR} determined $br(C_8, C_{2n})$. In this paper, we solve all remaining cases and give the exact values of $br(C_{2n}, C_{2m})$ for all $n\ge m\ge 5$, this answers a question concerned by Bucić-Letzter-Sudakov \cite{BLS}, Gholami-Rowshan \cite{GR}, Zhang-Sun \cite{Zhangs}, and Zhang-Sun-Wu \cite{Zhangsw}.

math.CO↗
arXiv2022

Lagrangian densities of some $3$-uniform hypergraphs

The Lagrangian density of an $r$-uniform hypergraph $H$ is $r!$ multiplying the supremum of the Lagrangians of all $H$-free $r$-uniform hypergraphs. For an $r$-uniform graph $H$ with $t$ vertices, it is clear that $π_λ(H)\ge r!λ{(K_{t-1}^r)}$. We say that an $r$-uniform hypergraph $H$ with $t$ vertices is $λ$-perfect if $π_λ(H)= r!λ{(K_{t-1}^r)}$. A theorem of Motzkin and Straus implies that all $2$-uniform graphs are $λ$-perfect. It is interesting to understand what kind of hypergraphs are $λ$-perfect. The property `$λ$-perfect' is monotone in the sense that an $r$-graph obtained by removing an edge from a $λ$-perfect $r$-graph (keep the same vertex set) is $λ$-perfect. It's interesting to understand the relation between the number of edges in a hypergraph and the `$λ$-perfect' property. We propose that the number of edges in a hypergraph no more than the number of edges in a linear hyperpath would guarantee the `$λ$-perfect' property. We show some partial result to support this conjecture. We also give some partial result to support the conjecture that the disjoint union of two $λ$-perfect $r$-uniform hypergraph is $λ$-perfect. We show that the disjoint union of a $λ$-perfect $3$-graph and $S_{2,t}=\{123,124,125,126,...,12(t+2)\}$ is perfect. This result implies the earlier result of Heftz and Keevash, Jiang, Peng and Wu, and several other earlier results.

math.CO↗
arXiv2021

An irrational Lagrangian density of a single hypergraph

The {\em Turán number} of an $r$-uniform graph $F$, denoted by $ex(n,F)$, is the maximum number of edges in an $F$-free $r$-uniform graph on $n$ vertices. The {\em Turán density} of $F$ is defined as $π(F)=\underset{n\rightarrow\infty}{\lim}{ex(n,F) \over {n \choose r }}.$ For graphs, Erdős-Stone-Simonovits (\cite{ESi}, \cite{ES}) showed that $Π_{\infty}^{(2)}=Π_{fin}^{(2)}=Π_{1}^{(2)}=\{0, {1 \over 2}, {2 \over 3}, \ldots,{l-1 \over l}, ...\}.$ We know quite few about the Turán density of an $r$-uniform graph for $r\ge 3$. Baber and Talbot \cite{BT}, and Pikhurko \cite{Pikhurko2} showed that there is an irrational number in $Π_{3}^{(3)}$ and $Π_{fin}^{(3)}$ respectively, disproving a conjecture of Chung and Graham \cite{FG}. Baber and Talbot \cite{BT} asked whether $Π_{1}^{(r)}$ contains an irrational number. In this paper, we show that the Lagrangian density of $F=\{123, 124, 134, 234, 567\}$ (the disjoint union of $K_4^3$ and an edge) is ${\sqrt 3\over 3}$, consequently, the Turán density of the extension of $F$ is an irrational number, answering the question of Baber and Talbot.

math.CO↗
arXiv2021

Non-jumping Turán densities of hypergraphs

A real number $α\in [0, 1)$ is a jump for an integer $r\ge 2$ if there exists $c>0$ such that no number in $(α, α+ c)$ can be the Turán density of a family of $r$-uniform graphs. A classical result of Erd\H os and Stone \cite{ES} implies that that every number in $[0, 1)$ is a jump for $r=2$. Erd\H os \cite{E64} also showed that every number in $[0, r!/r^r)$ is a jump for $r\ge 3$ and asked whether every number in $[0, 1)$ is a jump for $r\ge 3$. Frankl and Rödl \cite{FR84} gave a negative answer by showing a sequence of non-jumps for every $r\ge 3$. After this, Erd\H os modified the question to be whether $\frac{r!}{r^r}$ is a jump for $r\ge 3$? What's the smallest non-jump? Frankl, Peng, Rödl and Talbot \cite{FPRT} showed that ${5r!\over 2r^r}$ is a non-jump for $r\ge 3$. Baber and Talbot \cite{BT0} showed that every $α\in[0.2299, 0.2316)\cup [0.2871, \frac{8}{27})$ is a jump for $r=3$. Pikhurko \cite{Pikhurko2} showed that the set of all possible Turán densities of $r$-uniform graphs has cardinality of the continuum for $r\ge 3$. However, whether $\frac{r!}{r^r}$ is a jump for $r\ge 3$ remains open, and $\frac{5r!}{2r^r}$ has remained the known smallest non-jump for $r\ge 3$. In this paper, we give a smaller non-jump by showing that ${54r!\over 25r^r}$ is a non-jump for $r\ge 3$. Furthermore, we give infinitely many irrational non-jumps for every $r\ge 3$.

math.CO↗
arXiv2021

Tree Embeddings and Tree-Star Ramsey Numbers

We say that a graph $F$ can be embedded into a graph $G$ if $G$ contains an isomorphic copy of $F$ as a subgraph. Guo and Volkmann \cite{GV} conjectured that if $G$ is a connected graph with at least $n$ vertices and minimum degree at least $n-3$, then any tree with $n$ vertices and maximum degree at most $n-4$ can be embedded into $G$. In this paper, we give a result slightly stronger than this conjecture and obtain a sufficient and necessary condition that a tree with $n$ vertices and maximum degree at most $n-3$ can be embedded into a connected graph G with at least $n$ vertices and minimum degree at least $n-3$. Our result implies that the conjecture of Guo and Volkmann is true with one exception. We also give an application to the Ramsey number of a tree versus a star.

math.CO↗
arXiv2018

Lagrangian densities of hypergraph cycles

The Lagrangian density of an $r$-uniform hypergraph $F$ is $r!$ multiplying the supremum of the Lagrangians of all $F$-free $r$-uniform hypergraphs. For an $r$-graph $H$ with $t$ vertices, it is clear that $π_λ(H)\ge r!λ{(K_{t-1}^r)}$. We say that an $r$-unform hypergraph $H$ with $t$ vertices is perfect if $π_λ(H)= r!λ{(K_{t-1}^r)}$. A theorem of Motzkin-Straus implies that all $2$-uniform graphs are perfect. It is interesting to explore what kind of hypergraphs are perfect. A hypergraph is linear if any 2 edges have at most 1 vertex in common. We propose the following conjecture: (1) For $r\ge 3$, there exists $n$ such that a linear $r$-unofrm hypergraph with at least $n$ vertices is perfect. (2) For $r\ge 3$, there exists $n$ such that if $G, H$ are perfect $r$-uniform hypergraphs with at least $n$ vertices, then $G\bigsqcup H$ is perfect. Regarding this conjecture, we obtain a partial result: Let $S_{2,t}=\{123,124,125,126,...,12(t+2)\}$. (An earlier result of Sidorenko states that $S_{2,t}$ is perfect \cite{Sidorenko-89}.) Let $H$ be a perfect $3$-graph with $s$ vertices. Then $F=S_{2,t}\bigsqcup H$ is perfect if $s\geq 3$ and $t\geq 3$.

math.CO↗