arXiv subjects

Xuechun Zhang

Publications and source records attributed to Xuechun Zhang.

5 recordsLinked to original sources

arXiv2026

Stability for the Anti-Ramsey Number of Matchings

Let $n, r, s$ be three positive integers such that $n\geq 2s+5$. Let $K_r$ denote the complete graph of order $r$. Given a graph $F$, the anti-Ramsey number $ar(n,F)$ is defined as the minimum number $C$ such that any edge-coloring of $K_n$ with exactly $C$ colors contains a rainbow copy of $F$. Let $H$ be an edge-colored graph on $K_n$ with at least $g(n,s)$ colors, where \[ g(n,s)=\max\left\{ \binom{n}{2} - \binom{n - s + 1}{2} + 5, \binom{2s - 1}{2} + n + 1 \right\}. \] In this paper, we establish a stability type result for the anti-Ramsey number of matchings. Specifically, if $H$ does not have a rainbow matching of size $s+2$, then $H$ contains either a monochromatic complete graph $K_{n-s}$ or a monochromatic $K_{n - 2s - 1} \vee \overline{K_{2s + 1}}$.

math.CO↗

arXiv2025

On the Input-Output Monotonicity of Voltage Dynamics of Power System with Grid-Forming Converters

Integration of renewable resources is profoundly reshaping the dynamics of modern power systems. This study shows that the voltage dynamics of power systems with multiple grid-forming (GFM) converters often enjoys a desirable property called input-output monotonicity. A systematic approach for computing the derivatives of the voltage subsystem is presented first, which provides insight into the structural characteristics of these models. Next, the sign pattern of the trajectory Jacobian matrix associated with the voltage subsystem is analyzed and revealed. The analysis indicates that the voltage dynamics of power systems often exhibits the so-called input-output monotonicity property. The theoretical results are then validated through several simulation examples, underscoring their practical implications.

eess.SY↗

arXiv2023

A proof of Frankl-Kupavskii's conjecture on edge-union condition

A 3-graph $\mathcal{F}$ is \emph{$U(s, 2s+1)$} if for any $s$ edges $e_1,...,e_s\in E(\mathcal{F})$, $|e_1\cup...\cup e_s|\leq 2s+1$. Frankl and Kupavskii (2020) proposed the following conjecture: For any $3$-graph $\mathcal{F}$ with $n$ vertices, if $\mathcal{F}$ is $U(s, 2s+1)$, then $$e(\mathcal{F})\leq \max\left\{{n-1\choose 2}, (n-s-1){s+1\choose 2}+{s+1\choose 3}, {2s+1\choose 3}\right\}.$$ In this paper, we confirm Frankl and Kupavskii's conjecture.

math.CO↗

arXiv2021

Generalized Image Reconstruction over T-Algebra

Principal Component Analysis (PCA) is well known for its capability of dimension reduction and data compression. However, when using PCA for compressing/reconstructing images, images need to be recast to vectors. The vectorization of images makes some correlation constraints of neighboring pixels and spatial information lost. To deal with the drawbacks of the vectorizations adopted by PCA, we used small neighborhoods of each pixel to form compounded pixels and use a tensorial version of PCA, called TPCA (Tensorial Principal Component Analysis), to compress and reconstruct a compounded image of compounded pixels. Our experiments on public data show that TPCA compares favorably with PCA in compressing and reconstructing images. We also show in our experiments that the performance of TPCA increases when the order of compounded pixels increases.

cs.CV↗

arXiv2020

A characterization for graphs having strong parity factors

A graph $G$ has the \emph{strong parity property} if for every subset $X\subseteq V$ with $|X|$ even, $G$ has a spanning subgraph $F$ with minimum degree at least one such that $d_F(v)\equiv 1\pmod 2$ for all $v\in X$, $d_F(y)\equiv 0\pmod 2$ for all $y\in V(G)-X$. Bujtás, Jendrol and Tuza (On specific factors in graphs, \emph{Graphs and Combin.}, 36 (2020), 1391-1399.) introduced the concept and conjectured that every 2-edge-connected graph with minimum degree at least three has the strong parity property. In this paper, we give a characterization for graphs to have the strong parity property and construct a counterexample to disprove the conjecture proposed by Bujtás, Jendrol and Tuza.

math.CO↗