arXiv subjects

Junchi Zhang

Publications and source records attributed to Junchi Zhang.

17 recordsLinked to original sources

arXiv2026

Bounded Twin-Width Tournaments are $\dchi$-Bounded

For a tournament $T$ and a vertex ordering $\prec$, let $T^{\prec}$ be the graph of backward arcs in $\prec$. The diclique number of a tournament is $\domega(T)=\min_{\prec}\omega(T^{\prec})$, and the dichromatic number is $\dchi(T)=\min_{\prec}\chi(T^{\prec})$. We prove a mixed parameter transfer theorem: for all $k$ and $r$, if $\tww(T)\le k$ and $\omega(T^{\prec})\le r$, then the ordered twin-width of $(T^{\prec},\prec)$ is bounded by a function of $k$ and $r$. The proof combines the regular-semigrid theorem for ordered graphs with permutation-encoding obstructions to bounded twin-width in tournaments. Together with polynomial $\chi$-boundedness of graphs of bounded twin-width, this implies that tournaments of bounded twin-width are $\dchi$-bounded by $\domega$, resolving a conjecture of Aboulker, Aubian, Charbit, and Lopes.

math.CO↗

arXiv2026

Real-analytic finitely forcible kernels

Lov\'asz and Szegedy asked whether a nonconstant continuous, or even smooth, finitely forcible kernel exists. For every $0<\lambda\leq1/128$, we construct an explicit nonconstant real-analytic kernel $W_\lambda$ taking values in $(1/4,7/8)$. A single finite family of simple graphs, independent of $\lambda$, forces every $W_\lambda$ among all bounded symmetric real-valued kernels on $[0,1]^2$.

math.CO↗

arXiv2026

Infinitesimal finite forcibility and step kernels

We characterize infinitesimal finite forcibility for bounded symmetric real kernels. We prove that the graph-density gradients at a kernel span a finite-dimensional space if and only if the kernel is a step kernel. Combined with known finite-forcing results for step kernels, this gives a positive answer to a question of Lov\'asz and Szegedy on whether every infinitesimally finitely forcible kernel is finitely forcible. The proof combines spectral methods with a compression argument based on book graphs.

math.CO↗

arXiv2026

High-Energy Microresonator Soliton Generation

Kerr resonators generate stable frequency combs in a compact platform with applications in coherent communications, sensing, quantum information processing, and astrophysics. Ultrashort pulses can be generated, moreover, with a wavelength and repetition rate flexibility inaccessible by traditional mode-locked lasers, which is desirable for high peak-power applications including in biomedicine and materials processing. However, for these applications, single-pulse energies will need to be significantly improved beyond the fJ level characteristic of sources today. Here we describe and demonstrate a simple approach for increasing the pulse energy in anomalous dispersion Kerr microresonators. Through scaling laws based on a mean-field cavity model and supported by experimentally-accurate numerical simulations, we show that pulse energy scales strongly with output coupling if supported by sufficient drive power. In strongly over-coupled cavities, femtosecond pulses can be stabilized with pulse energy beyond the pJ level. With this theoretical framework, by pumping a 12 GHz Si3N4 cavity with 30% output coupling with 2.6-ps time-lens generated pulses, we observe stable 74-fs pulses with a record output pulse energy of 6 pJ. High energy Kerr resonators are anticipated to improve performance for current applications and compliment mode locked lasers for high peak power applications.

physics.optics↗

arXiv2026

The list coloring number of uncrowded hypergraphs

We prove that for every fixed integer $r\geq 2$ and every $\varepsilon>0$, every sufficiently large finite uncrowded $(r+1)$-uniform hypergraph of maximum degree $\Delta$ has list chromatic number at most \[ (1+\varepsilon)\left(\frac{r\Delta}{\log\Delta}\right)^{1/r}. \] The proof is a semi-random list-coloring nibble carried out directly on the original hypergraph. We encode the remaining coloring problem by active edge-color constraints and control all residual sizes through a binomial degree bound. After the nibble reaches a sparse terminal state, the coloring is completed by a Rosenfeld-style counting argument.

math.CO↗

arXiv2026

PhaseWin: An Efficient Search Algorithm for Faithful Visual Attribution

Visual attribution is a fundamental tool for interpreting modern vision and vision-language models, particularly when their decisions must be inspected, diagnosed, or audited. Its goal is to explain how a model's decision depends on local regions of the visual input, typically by assigning an importance ordering over candidate image regions. Given an image partitioned into $n$ regions, faithful attribution can be cast as an ordered subset-search problem, in which progressively inserting the selected regions should recover the target model response as early as possible. Exhaustive search over region subsets incurs exponential cost, while the widely used greedy search still requires a quadratic number of model evaluations, because every selection step rescores all remaining candidates. We propose PhaseWin, an efficient subset-search algorithm for faithful visual attribution. PhaseWin reorganizes greedy region selection into a phased window-search procedure: rather than re-evaluating the full candidate set at every step, it alternates between global candidate screening, adaptive pruning, and localized window refinement, while preserving the essential region-ranking behavior of greedy search. We analyze PhaseWin under monotone evidence-accumulation conditions and show that, under feature-level structural assumptions, it attains controllable linear evaluation complexity together with near-greedy faithfulness guarantees. Extensive experiments on image classification, object detection, visual grounding, and image captioning show that, among all compared attribution methods, PhaseWin reaches high faithfulness with the fewest forward passes, empirically realizing the predicted reduction from $O(n^2)$ to $O(n)$. The code is available at https://github.com/Qihuai27/phasewin-va.

cs.CV↗

arXiv2026

On independent sets in uncrowded uniform hypergraphs

We prove an average-degree lower bound on the independence number of uncrowded uniform hypergraphs. For every fixed $r\geq 2$ and every $\eta>0$, there exists $d_*=d_*(r,\eta)$ such that any uncrowded $(r+1)$-uniform hypergraph $G$ with $n$ vertices and average degree $d\geq d_*$ satisfies \[ \alpha(G)\geq (1-\eta)r^{-1/r}\left(\frac{\log d}{d}\right)^{1/r}n. \] The proof combines a cleaning procedure, which reduces the maximum top-layer degree to the average scale, with a random nibble procedure that repeatedly extracts independent vertices while controlling all lower-order degrees created by the process. After an initial top-layer cleaning, we run a trace nibble. Since the residual hypergraph contains traces of all sizes $2,\ldots,r+1$, we track the maximum degrees in every layer. A binomial-type recurrence for this degree profile yields the stated leading constant.

math.CO↗

arXiv2026

Submodular flows and extreme flows on measurable spaces

The theory of submodular flows, introduced by Edmonds and Giles, is a cornerstone of combinatorial optimization, unifying network flows, matroid intersections and directed cut coverings. In this paper, we establish a measurable-space version of this framework, addressing the structural existence and duality questions raised as part of Problem~10.6 by Lov\'asz in \textit{Submodular setfunctions on sigma-algebras, version 2}. We develop a theory of submodular flows on standard Borel spaces and establish the measurable analogues of the existence and optimality theorems. Furthermore, we introduce a measurable notion of the residual graph and characterize extreme flows by combining a base-polytope intersection condition with an acyclicity condition for the measurable residual graph, generalizing the discrete geometric intuition to the infinite-dimensional setting. Finally, we apply the theory to constrained supply-demand problems on measurable bipartite graphs and to fractional measurable orientations.

math.CO↗

arXiv2026

See Before You Code: Learning Visual Priors for Spatially Aware Educational Animation Generation

Large language models can generate executable code for educational animations, but the resulting renders often exhibit visual defects, including element overlap, misalignment, and broken animation continuity. These defects cannot be reliably detected from the code alone and become apparent only after execution. We formalize this problem as render-feedback-aware constrained code generation: given a natural language specification, the model must generate executable code whose rendered output satisfies structured quality criteria that can be evaluated only after rendering. To address this problem, we introduce OmniManim, a render-feedback-aware educational animation generation framework built around a shared scene state, explicit visual planning, structured post-render diagnostics, and localized repair. Within OmniManim, the Vision Agent is a task-specific visual planning module: it predicts sparse keyframe layouts with coarse-to-fine bounding-box denoising and optimizes an interpolation-aware objective to reduce intermediate-frame failures induced by downstream animation interpolation. We further construct two datasets, ManimLayout-1K and EduRequire-500, and provide a reproducible evaluation protocol covering executability, instructional quality, visual quality, and efficiency. On EduRequire-500, OmniManim improves measured render quality over both single-model baselines and existing multi-agent frameworks. Systematic ablation studies further verify that explicit visual planning, especially its coarse spatial prior, bounding-box refinement, and interpolation-aware optimization, is central to these gains.

cs.AI↗

arXiv2026

Hypergraph independence bounds: from maximum degree to average degree

We prove a transfer theorem for hereditary classes of $(r+1)$-uniform hypergraphs. Let $\mathcal H$ be such a class, and for $H\in\mathcal H$ write $\Delta(H)$ and $d(H)$ for the maximum degree and average degree of $H$, respectively. We show that, for every nearly logarithmic function $f$ in the sense defined below, a maximum-degree lower bound for the independence number of the form \[ \alpha(H)\ge (1-o(1))\frac{f(\Delta(H))}{\Delta(H)^{1/r}}|V(H)| \qquad\text{as }\Delta(H)\to\infty \] for all $H\in\mathcal H$ implies the corresponding average-degree lower bound \[ \alpha(H)\ge (1-o(1))\frac{f(d(H))}{d(H)^{1/r}}|V(H)| \qquad\text{as }d(H)\to\infty . \] We combine this transfer theorem with known coloring and fractional-coloring bounds to obtain consequences for graphs excluding a fixed cycle, graphs with bounded clique number, locally $q$-colorable graphs, and locally sparse uniform hypergraphs.

math.CO↗

arXiv2026

The linear Tur\'an number of the 3-graph $P_5$

We prove that for any linear 3-graph on $n$ vertices without a path of length 5, the number of edges is at most $\frac{15}{11}n$, and the equality holds if and only if the graph is the disjoint union of $G_0$, a graph with 11 vertices and 15 edges. Thus, $ex_L(n,P_5)\leq \frac{15}{11}n$, and the equality holds if and only if $11|n$.

math.CO↗

arXiv2025

Flag-transitive $2$-$(v,k,\lambda)$ designs with $\lambda\ge (r,\lambda)^2$

This paper is devoted to the study of $2$-designs with $\lambda\ge (r,\lambda)^2$ admitting a flag-transitive automorphism group $G$. The group $G$ has been shown to be point-primitive of either almost simple or affine type. In this paper, we classify the $2$-designs with $\lambda \geq (r,\lambda)^2>1$ admitting a flag-transitive almost simple automorphism group with socle $\mathrm{PSL}_n(q)$ or $\mathrm{PSU}_n(q)$ for $n \geq 3$.

math.CO↗

arXiv2025

PhaseWin Search Framework Enable Efficient Object-Level Interpretation

Attribution is essential for interpreting object-level foundation models. Recent methods based on submodular subset selection have achieved high faithfulness, but their efficiency limitations hinder practical deployment in real-world scenarios. To address this, we propose PhaseWin, a novel phase-window search algorithm that enables faithful region attribution with near-linear complexity. PhaseWin replaces traditional quadratic-cost greedy selection with a phased coarse-to-fine search, combining adaptive pruning, windowed fine-grained selection, and dynamic supervision mechanisms to closely approximate greedy behavior while dramatically reducing model evaluations. Theoretically, PhaseWin retains near-greedy approximation guarantees under mild monotone submodular assumptions. Empirically, PhaseWin achieves over 95% of greedy attribution faithfulness using only 20% of the computational budget, and consistently outperforms other attribution baselines across object detection and visual grounding tasks with Grounding DINO and Florence-2. PhaseWin establishes a new state of the art in scalable, high-faithfulness attribution for object-level multimodal models.

cs.CV↗

arXiv2025

New lower bounds on the non-repetitive chromatic number of some graphs

A graph $ G $ is said to be (vertex) non-repetitively colored if no simple path in $ G $ has a sequence of vertex colors that forms a repetition. Formally, a coloring $ c: V(G) \to \{1, 2, \dots, k\} $ is non-repetitive if, for every path $\langle v_1, v_2, \dots, v_{2m} \rangle$ in $ G $, the sequence of colors $ c(v_1), c(v_2), \dots, c(v_{2m}) $ is not of the form $ ww $, where $ w $ is a sequence of $ m $ colors. The minimum number of colors required for such a coloring is called the \emph{non-repetitive chromatic number} of $G$, denoted by $\pi(G)$. In this paper, we primarily prove that $\pi(P \square P) \ge 6$ and $\pi(P \boxtimes P) \ge 9$, where $ P \square P $ and $ P \boxtimes P $ are the Cartesian product and the strong product of two infinite paths, respectively. This improves upon the previous best lower bounds.

math.CO↗

arXiv2025

Unbounded degree spanning hypertrees in Dirac hypergraphs

In 2001, Koml\'os, S\'ark\"ozy, and Szemer\'edi proved that every sufficiently large $n$-vertex graph with minimum degree at least $\left(1/2+\gamma\right)n$ contains all spanning trees with maximum degree at most $cn/\log n$. We extend this result to hypergraphs by considering loose hypertrees, which are linear hypergraphs obtained by successively adding edges that share exactly one vertex with a previous edge. For all $k > \ell \geq 2$, we determine asymptotically optimal $\ell$-degree conditions that ensure the existence of all rooted spanning loose hypertrees, without any degree condition, in terms of the $(\ell-1)$-degree threshold for the existence of a perfect matching in $(k-1)$-graphs. As a corollary, we also asymptotically determine the $\ell$-degree threshold for the existence of bounded degree spanning loose hypertrees in $k$-graphs for $k/2 < \ell < k$, confirming a conjecture of Pehova and Petrova in this range. In our proof, we avoid the use of Szemer\'edi's regularity lemma.

math.CO↗

arXiv2025

Limit for Quotient Convergent Graph Sequence

Building on the limit theory for set functions, we prove that the limit of convergent sequence of bounded-degree graphs' cycle matroids can be represented as the cycle matroid of a graphing, analogous to the completeness result for local-global convergence.

math.CO↗

arXiv2025

On 1-11-representability and multi-1-11-representability of graphs

Jeff Remmel introduced the concept of a $k$-11-representable graph in 2017. This concept was first explored by Cheon et al. in 2019, who considered it as a natural extension of word-representable graphs, which are exactly 0-11-representable graphs. A graph $G$ is $k$-11-representable if it can be represented by a word $w$ such that for any edge (resp., non-edge) $xy$ in $G$ the subsequence of $w$ formed by $x$ and $y$ contains at most $k$ (resp., at least $k+1$) pairs of consecutive equal letters. A remarkable result of Cheon at al. is that any graph is 2-11-representable, while it is still unknown whether every graph is 1-11-representable. Cheon et al. showed that the class of 1-11-representable graphs is strictly larger than that of word-representable graphs, and they introduced a useful toolbox to study 1-11-representable graphs, which was extended by additional powerful tools suggested by Futorny et al. in 2024. In this paper, we prove that all graphs on at most 8 vertices are 1-11-representable hence extending the known fact that all graphs on at most 7 vertices are 1-11-representable. Also, we discuss applications of our main result in the study of multi-1-11-representation of graphs we introduce in this paper analogously to the notion of multi-word-representation of graphs suggested by Kenkireth and Malhotra in 2023.

math.CO↗