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Leilei Zhang

Publications and source records attributed to Leilei Zhang.

At least 19 recordsLinked to original sources
arXiv2026

On the distinct maximal-clique sizes in $k$-uniform hypergraphs

Let $g(n,k)$ be the maximum number of distinct sizes of maximal cliques in an $n$-vertex $k$-uniform hypergraph, and let $f(n,k)=n-g(n,k)$. We determine the asymptotic order of $f(n,k)$ for every fixed integer $k\ge 3$. Define $L_2(x)=\max\{2,\log_2(\max\{1,x\})\}$, and, for $j\ge 3$, let $L_j(x)$ be the least number of iterations of $L_{j-1}$ needed to reach a value at most $16$. We prove that $$ f(n,k)=\Theta_k(L_k(n)).$$ In particular, $f(n,3)=\Theta(\log^{*}n)$, where $\log^{*}n$ denotes the iterated logarithm. We also determine the asymptotic behaviour of the layered-tree threshold $c(n,k)$ arising from Gao's insertion-tree method: $$ c(n,k)=\log_2 L_k(n)+O_k(1). $$ Consequently, $f(n,k)=\Theta_k\!\left(2^{c(n,k)}\right)$. Our result gives a negative answer to Gao's question in the case $k=3$.

math.CO↗
arXiv2026

Counterexamples to the Balogh-Linz-Patk\'os Conjecture

A set system $\mathcal{F}$ is called $t$-intersecting if $|A\cap B|\ge t$ for every pair of sets $A,B\in \mathcal{F}.$ A set system $\mathcal{F}$ is $k$-Sperner if it does not contain a chain of length $k+1$. Balogh, Linz and Patk\'os (Combinatorial Theory, 2023) conjectured an extremal result for $t$-intersecting $k$-Sperner families when $n+t$ is odd. In this note we give an explicit construction that is $t$-intersecting and $k$-Sperner, and whose size exceeds that of the conjectured fixed-star construction for infinitely many values of $n$. Consequently, we disprove the Balogh-Linz-Patk\'os conjecture for all $t\ge 2$ and $k\ge 2$ satisfying $k(t-1)\ge t+1$.

math.CO↗
arXiv2026

Further Results on the Maximum Number of Stars in Graphs with Forbidden Properties

A graph $G$ is called $k$-edge-hamiltonian if every linear forest (i.e., a disjoint union of paths) with at most $k$ edges is contained in a Hamilton cycle of $G$. In 2018, F\"uredi, Kostochka and Luo determined the maximum number of $t$-stars in nonhamiltonian graphs, thereby extending an earlier result of Erd\H{o}s. Recently, Berikkyzy, Hogenson, Kirsch and McDonald extended this line of research by determining the maximum number of $t$-stars in graphs that are not $k$-edge-hamiltonian, as well as in graphs failing to satisfy related properties such as traceability, hamiltonian-connectedness and $k$-hamiltonicity. For sufficiently large $t$, they also characterized the extremal graphs, while for smaller values of $t$, they proposed a conjecture. In this paper, we investigate this conjecture. We show that the conjecture fails at the critical value and further establish a threshold-type result describing the behavior of the extremal graphs when $t$ is close to this critical value.

math.CO↗
arXiv2026

The maximum number of triangles in graphs without vertex disjoint friendship graphs

Given graphs $H$ and $F$, the generalized Tur\'an number $\mathrm{ex}(n,H,F)$ is the maximum number of copies of $H$ among all $n$-vertex $F$-free graphs. The friendship graph $F_k$ consists of $k$ triangles sharing a common vertex. In this paper, we determine the value of $\mathrm{ex}(n,K_3,(t+1)F_k)$, where $K_3$ is a triangle, $t\geq 1$ is an integer, and $(t+1)F_k$ denotes a union of $(t+1)$ pairwise vertex-disjoint copies of $F_k$. Moreover, we characterize the extremal structure. Our result can be viewed as a generalization of the result of Zhu, Chen, Gerbner, Gy\H{o}ri, and Hama Karim, as well as of the remaining case left open by Wang, Ni, Liu, and Kang. In contrast to the extremal graphs of $F_k$, the extremal graphs of $(t+1)F_k$ undergo a fundamental change. This structure is also different from those of previous similar problems.

math.CO↗
arXiv2026

Hamiltonian Properties of 3-Connected Claw-Free Graphs and Line Graphs of 3-Hypergraphs

Motivated by Thomassen's well-known line graph conjecture, many researchers have explored sufficient conditions for claw-free graphs to be Hamiltonian or Hamilton-connected. In 1994, Ageev proved that every $2$-connected claw-free graph with domination number at most $2$ is Hamiltonian. In this paper, we extend this line of research to $3$-connected graphs by establishing the best possible upper bound on the domination number that guarantees Hamiltonicity. Specifically, we show that, except for some well-defined exceptional graphs, every $3$-connected claw-free graph $G$ with domination number at most $5$ is Hamiltonian. Furthermore, we prove that, apart from a few exceptional cases, every $3$-connected claw-free graph $G$ with domination number at most $4$ is Hamilton-connected, thereby generalizing earlier results of Zheng, Broersma, Wang and Zhang and Vr\'ana, Zhan and Zhang. We further investigate the Hamiltonian properties of line graphs of $3$-hypergraphs, and prove that every 3-connected line graph of a 3-hypergraph with domination number at most $4$ is Hamiltonian.

math.CO↗
arXiv2026

A Note on Gr\"{u}nbaum's Conjecture about Longest Cycles and Paths

Let $c(G)$ denote the circumference of a graph $G$, i.e., the number of vertices in its longest cycle. For positive integers $n$ and $k$ with $n>k$, let $\varGamma(n;k)$ be the class of graphs of order $n$ with $c(G) = n-k$ such that every induced subgraph of order $n-k$ is Hamiltonian. When $k=$, the class $\varGamma(n; 1)$ coincides with the family of hypohamiltonian graphs-non-Hamiltonian graphs in which the deletion of any single vertex yields a Hamiltonian graph.Replacing Hamiltonian with traceable and $c(G)$ with $p(G)$, the order of a longest path, defines the analogous class $\varPi(n;k)$.Gr\"{u}nbaum (1974) conjectured that both $\varGamma(n; k)$ and $\varPi(n; k)$ are empty for all $n>k \ge 2$. In this note, we first establish upper bounds on the maximum degree of graphs in the classes $\varGamma(n; k)$ and $\varPi(n; k)$. Using these bounds, we show that $\varGamma(n; k)$ is empty when $n<k^2+2k+3$, and that $\varPi(n; k)$ is empty when $n<k^2+2k+2$. These results provide further evidence supporting Gr\"{u}nbaum's conjecture.

math.CO↗
arXiv2026

Stability results for Berge-matching in hypergraphs

Given a graph $F$, a hypergraph is called a Berge-$F$ if it can be obtained by expanding each edge of $F$ into a hyperedge containing it. Let $M_{k}$ denote the matching of size $k$. Kang, Ni, and Shan [12] determined the Tur\'an number of Berge-$M_k$. Our main result shows that if an $r$-uniform hypergraph $H$ on $n$ vertices has nearly as many edges as the extremal in their theorem without containing $M_k$, then $H$ must be structurally close to certain well-specified graphs. Meanwhile, our result also implies several stability results, such as the stability version of the well-known Erd\H{o}s-Gallai theorem (Erd\H{o}s and Gallai, 1959 [5]).

math.CO↗
arXiv2025

AgentBay: A Hybrid Interaction Sandbox for Seamless Human-AI Intervention in Agentic Systems

The rapid advancement of Large Language Models (LLMs) is catalyzing a shift towards autonomous AI Agents capable of executing complex, multi-step tasks. However, these agents remain brittle when faced with real-world exceptions, making Human-in-the-Loop (HITL) supervision essential for mission-critical applications. In this paper, we present AgentBay, a novel sandbox service designed from the ground up for hybrid interaction. AgentBay provides secure, isolated execution environments spanning Windows, Linux, Android, Web Browsers, and Code interpreters. Its core contribution is a unified session accessible via a hybrid control interface: An AI agent can interact programmatically via mainstream interfaces (MCP, Open Source SDK), while a human operator can, at any moment, seamlessly take over full manual control. This seamless intervention is enabled by Adaptive Streaming Protocol (ASP). Unlike traditional VNC/RDP, ASP is specifically engineered for this hybrid use case, delivering an ultra-low-latency, smoother user experience that remains resilient even in weak network environments. It achieves this by dynamically blending command-based and video-based streaming, adapting its encoding strategy based on network conditions and the current controller (AI or human). Our evaluation demonstrates strong results in security, performance, and task completion rates. In a benchmark of complex tasks, the AgentBay (Agent + Human) model achieved more than 48% success rate improvement. Furthermore, our ASP protocol reduces bandwidth consumption by up to 50% compared to standard RDP, and in end-to-end latency with around 5% reduction, especially under poor network conditions. We posit that AgentBay provides a foundational primitive for building the next generation of reliable, human-supervised autonomous systems.

cs.AI↗
arXiv2025

Synergizing Large Language Models and Task-specific Models for Time Series Anomaly Detection

In anomaly detection, methods based on large language models (LLMs) can incorporate expert knowledge by reading professional document, while task-specific small models excel at extracting normal data patterns and detecting value fluctuations from training data of target applications. Inspired by the human nervous system, where the brain stores expert knowledge and the peripheral nervous system and spinal cord handle specific tasks like withdrawal and knee-jerk reflexes, we propose CoLLaTe, a framework designed to facilitate collaboration between LLMs and task-specific models, leveraging the strengths of both models for anomaly detection. In particular, we first formulate the collaboration process and identify two key challenges in the collaboration: (1) the misalignment between the expression domains of the LLMs and task-specific small models, and (2) error accumulation arising from the predictions of both models. To address these challenges, we then introduce two key components in CoLLaTe: a model alignment module and a collaborative loss function. Through theoretical analysis and experimental validation, we demonstrate that these components effectively mitigate the identified challenges and achieve better performance than both LLM-based and task-specific models.

cs.AI↗
arXiv2024

The maximum size of a nonhamiltonian-connected graph with given order and minimum degree

In this paper, we determine the maximum size of a nonhamiltonian-connected graph with prescribed order and minimum degree. We also characterize the extremal graphs that attain this maximum size. This work generalizes a previous result obtained by Ore [ J. Math. Pures Appl. 42 (1963) 21-27] and further extends a theorem proved by Ho, Lin, Tan, Hsu, and Hsu [Appl. Math. Lett. 23 (2010) 26-29]. As a corollary of our main result, we determine the maximum size of a $k$-connected nonhamiltonian-connected graph with a given order.

math.CO↗
arXiv2024

Theoretical Study on the Structural and Thermodynamic Properties of U-He compounds under High Pressure

Uranium is considered as a very important nuclear energy material because of the huge amount of energy released. As the main products of spontaneous decay of uranium, helium is difficult to react with uranium for its chemical inertness. Therefore, bubbles will be formed inside uranium, which could greatly reduce the performance of uranium or cause the safety problems. Additionally, nuclear materials are usually operated in an environment of high-temperature and high-pressure, so it is necessary to figure out the exact state of helium inside uranium at extreme conditions. Here, we explored the structural stability of U-He system under high-pressure and high-temperature by using density functional theory calculations. Two metastable phases are found between 50 and 400 GPa: U4He with space group Fmmm and U6He with space group P-1. Both are metallic and adopt layered structures. Electron localization function calculation combined with charge density difference analysis indicate that there are covalent bonds between U and U atoms in both Fmmm-U4He and P-1-U6He. Compared with the elastic modulus of $α$-U, the addition of helium has certain influence on the mechanical properties of uranium. Besides, first-principles molecular dynamics simulations were carried out to study the dynamical behavior of Fmmm-U4He and P-1-U6He at high-temperature. It is found that Fmmm-U4He and P-1-U6He undergo one-dimensional superionic phase transitions at 150 GPa. Our study revealed exotic structure of U-He compounds beyond the form of bubble under high-pressure and high-temperature, that might be relevant to the performance and safety issue of nuclear materials at extreme conditions.

cond-mat.mtrl-sci↗
arXiv2024

Extremal problems about the order and size of nonhamiltonian locally linear graphs

The relation between local structure and global cycle properties is a classical topic in graph theory. A graph $G$ is locally linear if $G[N(v)]$ is a path for every $v\in V(G)$. It is locally Hamiltonian or locally traceable if every vertex neighborhood induces a Hamiltonian or traceable graph, respectively. Earlier work by Pareek and Skupie\'{n}, Skupie\'{n}, Davies and Thomassen, Asratian and Oksimets, and de Wet and van Aardt studied extremal questions for these related graph classes. We prove that the minimum order of a nonhamiltonian locally linear graph is $12$ and that, for every integer $n\geq 12$, the minimum size of such a graph of order $n$ is $2n$. We also prove that every nontraceable locally linear graph of order $n$ has at least $2n+3$ edges.

math.CO↗
arXiv2024

Universal Metallic Surface States in Electride

Robust metallic surface states (MSS) of topological insulator (TI) against imperfections and perturbations are important in broad applications such as chemical catalysis and quantum computing. Unfortunately, they are suffered from the narrow band gap that can be accessed. Searching for MSS with large bulk band gap beyond conventional TIs becomes a quest. In this work, inspired by the adiabatic connection principle in real space, we identify that all electrides, a new class of emerging materials, must host robust and universal MSS that resists any disturbances, in spite of the fact that some of them could be classified as trivial in standard topology theory. This counterintuitive property is traced to the specific charge localization-delocalization change intrinsic to electride when approaching the crystalline surface or interface, which is a kind of interstice-centered to atom-centered transition in the real-space topology of the charge density distribution, and is sharply different from the band inversion in the standard topology theory. The new mechanism circumvents the obstacle that limits the band gap of TI. Robust and universal MSS in an electride that conventionally-determined as trivial but with a colossal band gap beyond 6.13 eV are demonstrated. This gap size is about 6-fold larger than the highest record of known "wide-gap" TIs, thus opens up new avenues to universal MSS with gigantic bulk gap.

cond-mat.mtrl-sci↗
arXiv2023

Further results on the number of cliques in graphs covered by long cycles

Let $Γ(n,k)$ be the set of $2$-connected $n$-vertex graphs containing an edge that is not on any cycle of length at least $k+1.$ Let $g_s(n,k)$ denote the maximum number of $s$-cliques in a graph in $Γ(n,k).$ Recently, Ji and Ye [SIAM J. Discrete Math., 37 (2023) 917-924] determined $g_s(n,k).$ They remark that it is interesting to characterize the extremal graphs. In this paper, we give such a characterization.

math.CO↗
arXiv2022

Graphs with many independent vertex cuts

The cycles are the only $2$-connected graphs in which any two nonadjacent vertices form a vertex cut. We generalize this fact by proving that for every integer $k\ge 3$ there exists a unique graph $G$ satisfying the following conditions: (1) $G$ is $k$-connected; (2) the independence number of $G$ is greater than $k;$ (3) any independent set of cardinality $k$ is a vertex cut of $G.$ The edge version of this result does not hold. We also consider the problem when replacing independent sets by the periphery.

math.CO↗
arXiv2022

The minimum Kirchhoff index of phenylene chains

Let $G$ be a connected graph. The resistance distance between any two vertices of $G$ is equal to the effective resistance between them in the corresponding electrical network constructed from $G$ by replacing each edge with a unit resistor. The Kirchhoff index is defined as the sum of resistance distances between all pairs of the vertices. Recently, Yang and Wang determined the maximum Kirchhoff index of phenylene chains, and they proposed a conjecture about the minimum Kirchhoff index. In this note, we characterized the minimum phenylene chains with respect to the Kirchhoff index. This proves the conjecture.

math.CO↗