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Jin Yan

Publications and source records attributed to Jin Yan.

At least 19 recordsLinked to original sources
arXiv2026

Sharp digirth bounds for C(k,k)-subdivisions in digraphs

For an integer $k\ge2$, let $C(k,k)$ denote the oriented cycle formed by two internally vertex-disjoint directed paths of length $k$ with the same initial and terminal vertices. We extend a problem of Picasarri-Arrieta and Rambaud [European J. Combin. 122 (2024)] on subdivisions of $C(k,k)$ to digraphs with several vertices of out-degree one. Following the sharp result of Chen et al. [arXiv:2610.38574v1 (2026)] for at most one such vertex, we determine the exact digirth threshold when at most $r\ge4$ such vertices are permitted. Specifically, for all integers $k\ge2$ and $r\ge4$, every digraph of minimum out-degree at least one, with at most $r$ vertices of out-degree one and digirth at least $r(k-1)+1$, contains a subdivision of $C(k,k)$.

math.CO↗
arXiv2026

$K_r$-Packings in Graphs with $t_{r-1}(n)+k$ Edges

Fix an integer $r\ge 3$. Let $K_r$ denote the complete graph on $r$ vertices and $t_{r-1}(n)$ the Turán number for $K_r$. For each integer $k$ satisfying $0\le k \le \binom{n}{2}-t_{r-1}(n)$, $\rev{ϕ_r(n,k)}$ denotes the minimum, over all $n$-vertex graphs with $t_{r-1}(n)+k$ edges, of the maximum size of a $K_r$-packing \rev{(a family of edge-disjoint copies of $K_r$)} in the graph. The problem of characterizing the behavior of $ϕ_r(n,k)$ has a long history and dates back to Erdős for the case $r=3$. \rev{Győri} proved that $ϕ_r(n,k)=(1-o(1))k$ whenever $k=o(n^2)$. We characterize the range of $k$ for which $ϕ_r(n,k)=(1-o(1))k$, showing that this equality holds if and only if $k=o(n^2)$. Moreover, we obtain an \rev{asymptotic} upper bound (which falls below $1$) for $ϕ_r(n,k)/k$ whenever $k/n^2$ converges to a constant $δ$ satisfying $0 < δ\le \frac{1}{2(r-1)}$.

math.CO↗
arXiv2026

Complexity of Arc-Decompositions involving Perfect Matchings and Cycle Factors

For two digraph properties $P_1$ and $P_2$, a $(P_1,P_2)$-arc-decomposition of a digraph $D$ is a partition $A(D)=A_1\mathbin{\dot\cup}A_2$ such that the spanning subdigraphs $D[A_1]$ and $D[A_2]$ have properties $P_1$ and $P_2$, respectively. We prove that it is NP-complete to decide whether a digraph admits an arc-decomposition with properties $(P_1,P_2)$ where $(P_1,P_2)\in \{$(perfect matching, having no odd cycle), (perfect matching, strong), (perfect matching, having an out-branching), (cycle factor, having no odd cycle)$\}$. All four problems remain NP-complete even when restricted to strongly connected digraphs. We also prove that every strongly connected digraph with minimum semi-degree at least three contains two vertex-disjoint cycles whose simultaneous arc deletion leaves the digraph strongly connected. These results settle some open problems posed by Bang-Jensen, Bessy, Gonçalves, and Picasarri-Arrieta [Theoret. Comput. Sci. 928 (2022), 167--182].

math.CO↗
arXiv2026

The unilateral 2-linkage problem

We study a new variant of the classical linkage problem for digraphs, which we call {\bf unilateral linkages.} A digraph $D$ is {\bf unilaterally $\bm{k}$-linked} if it satisfies that for every choice of $2k$ distinct vertices $s_1,s_2,\ldots{},s_k,t_1,t_2,\ldots{},t_k$ there are vertex disjoint directed paths $P_1,\ldots{},P_k$ in $D$ such that $P_i$ is either an $(s_i,t_i)$-path or a $(t_i,s_i)$-path for each $i\in [k]$. This is a relaxation of the property of $D$ being $k$-linked, where we insist that each $P_i$ must go from $s_i$ to $t_i$. Clearly every $k$-linked digraph is also unilaterally $k$-linked. It is easy to see that the unilateral linkage problem is NP-complete for general digraphs and that it is trivial for semicomplete digraphs (digraphs with no pair of non-adjacent vertices). We show how Thomassen's construction of highly connected digraphs which are not 2-linked can be modified to show that there is no connectivity which guarantees that a digraph is unilaterally 2-linked. We show that every 4-strong split digraph and every 5-strong semicomplete multipartite digraph is unilaterally 2-linked. Both bounds are best possible. A digraph is {\bf co-bipartite} if its vertex set can be covered by two induced semicomplete digraphs. We prove that every 6-strong co-bipartite digraph is unilaterally 2-linked. Here we do not know whether the correct answer is 5. We also prove that every 16-strong digraph with independence number at most 2 is unilaterally 2-linked.

math.CO↗
arXiv2026

Spanning Subdivisions in Robustly Expanding Digraphs

Kühn, Osthus and Treglown [\emph{J. Combin. Theory Ser. B} \textbf{100} (2010)] proved that every sufficiently large robust outexpander with linear minimum semi-degree contains a Hamilton cycle. In this paper, we extend their result to spanning subdivisions of a digraph $F$ in a robust outexpander $D$, with prescribed branch vertices and arbitrarily specified path lengths. In particular, the digraph $F$ need not be fixed. For an $n$-vertex digraph $D$, $F$ may have as many as a constant multiple of $n/\log n$ arcs. This holds provided that $F$ has sufficiently few vertices and the prescribed lengths satisfy suitable conditions. We further establish a more general result for mixed prescribed path lengths, which allows $F$ to have linearly many arcs when the shorter replacement paths together use only a small proportion of the vertices of $D$. As applications, we obtain spanning subdivision extensions of the asymptotic Nash--Williams theorem for digraphs and an asymptotically sharp Ore-type theorem for oriented graphs.

math.CO↗
arXiv2026

Compiling Autoregressive LLM Control into Policy-Exact Parallel Inference for Hierarchical RAG

Large language models increasingly act as controllers in structured decision pipelines, yet their decisions are still executed by autoregressive text decoding. Hierarchical RAG is one such setting: at each frontier of a document tree, an LLM commits to $b$ of $k$ sections under an input-dependent action grammar, executed as either failure-prone free-form JSON or reliable but strictly serial grammar-constrained generation. We show that this serialization is unnecessary for bounded-horizon control: the reachable future forms a shallow action-prefix tree, which we compile --- one root prefill plus one tree-causal probe evaluate every prefix-conditioned action distribution with serial decoding's tokens, positions, and causal ancestry. A prefix-state lemma makes compiled control policy-exact in exact arithmetic: the same policy with a two-forward critical path and zero output tokens. One further probe pass certifies the exact order-marginalized MAP set. Across four document-disjoint splits and two scales on the primary MetaX device, compiled control matches the serial policy on $84$--$89\%$ of questions --- every instrumented divergence a floating-point near-tie --- and cuts control latency by $35$--$47\%$ versus token-wise serial decoding and $6$--$7\%$ versus an idealized chunked engine; an H800 repetition preserves the ordering.

cs.CL↗
arXiv2026

Boundary regularity of $k$-Hessian eigenfunctions

We prove smoothness up to the boundary for $k$-Hessian eigenfunctions on smooth bounded uniformly convex domains when $2\leq k<n$. This resolves an open problem in the regularity theory of degenerate fully nonlinear elliptic equations and extends the boundary smoothness theorem for Monge--Ampère eigenfunctions. In contrast to the Monge--Ampère case, the linearized $k$-Hessian operator remains uniformly elliptic up to the boundary.

math.AP↗
arXiv2026

Sharp Hessian integrability for the planar Monge--Ampère equation

We determine the optimal local integrability of the Hessian for planar convex Alexandrov solutions satisfying $$ λ\,dx\leqμ_u\leqλ^{-1}\,dx, \qquad 0<λ<1. $$ The distributional Hessian satisfies the sharp endpoint estimate $$ D^2u\in L_{\mathrm{loc}}^{P,\infty}, \qquad P=\frac{1+λ}{1-λ}. $$ This settles the weak estimate proposed by De Philippis and Figalli and identifies the larger optimal exponent $P$. We further construct strictly convex $C^1$ solutions for which the distribution function of the Laplacian is comparable to $T^{-P}$ for all sufficiently large $T$. Hence both the exponent and the weak Lorentz endpoint are optimal: $L^{P,\infty}$ cannot be replaced by $L^{P,b}$ for any finite $b>0$. As applications, we obtain sharp estimates in Minty coordinates and optimal quantitative bi-Lipschitz restrictions for gradient maps, including an optimal exceptional-mass bound in quadratic-cost transport.

math.AP↗
arXiv2026

The Hajnal-Szemerédi theorem in digraphs revisited

The Hajnal--Szemerédi theorem states that every graph with $Δ(G)\le k-1$ admits an equitable $k$-coloring. Using a reduction implicit in Aboulker, Oijid, Petit, Rocton, and Simon (which itself builds on the literature surrounding feedback arc sets), we prove that this theorem is equivalent to each of four digraph analogues guaranteeing equitable acyclic $k$-colorings under various degree conditions. A minor modification of the proof method in makes the construction algorithmic, so the equitable acyclic coloring guaranteed by these statements can be found in polynomial time. The same reduction shows that the Ore-type Hajnal--Szemerédi theorem is equivalent to its four digraph analogues. These equivalences settle the conjecture of Treglown that every $n$-vertex digraph with $r\mid n$ and $δ^{\max}(D)\ge(1-\frac{1}{r})n$ contains a $T_r$-factor, answer a question of Molla and Treglown by determining the exact Ore-type threshold for $T_r$-factors, and recover the directed Hajnal--Szemerédi theorems of Czygrinow, DeBiasio, Kierstead, and Molla as immediate corollaries of this reduction. We also discuss related problems for oriented graphs which do not follow immediately from the reduction mentioned above.

math.CO↗
arXiv2026

An Ore-type Theorem for Oriented Discrepancy of Hamilton Cycles

Oriented discrepancy concerns the existence of prescribed subgraphs whose edges exhibit a substantial imbalance between the two directions. The Hamilton cycle case was first investigated by Gishboliner, Krivelevich, and Michaeli. Subsequently, Freschi and Lo established a sharp minimum-degree result and raised the question of whether a corresponding Ore-type theorem holds. For an oriented graph $G$ on $n$ vertices, define $σ_2(G):=\min\{d_G(x)+d_G(y):x\neq y,\ xy,yx\notin E(G)\},$ with $σ_2(G):=2(n-1)$ when $G$ is a tournament. Given a cycle $C$ in an oriented graph, let $σ_{\max}(C)$ denote the maximum number of edges oriented consistently with one of the two traversal directions of $C$. We prove that for every $γ>0$ there exists $n_0$ such that every oriented graph $G$ on $n\ge n_0$ vertices with $σ_2(G)\ge n$ contains a Hamilton cycle $C$ satisfying $$ σ_{\max}(C)\geq \max\left\{\frac n2,\frac{σ_2(G)}2-γn\right\}. $$ This partially answers a question of Freschi and Lo and gives an asymptotic affirmative answer to a conjecture of Ai et al. The bound is asymptotically best possible.

math.CO↗
arXiv2026

Strain-engineered long-distance supercurrent in an altermagnetic CrSb based-Josephson junction

Altermagnetic materials, characterized by vanishing net magnetization and momentum-dependent spin splitting, provide a unique platform for realizing unconventional superconducting states without the detrimental stray fields. Here, we demonstrate long-distance Josephson supercurrent transport through epitaxial thin films of the strained altermagnet CrSb. We observe a dissipationless supercurrent across a lateral spacer distance of 150 nm in a CrSb-based Josephson junction, exceeding the conventional spin-singlet coherence length by nearly two orders of magnitude. This supercurrent is further accompanied by a well-defined magnetic-field-induced quantum interference pattern, providing direct evidence for genuine Josephson coupling. Our results reveal altermagnets as a promising route toward generating long-distance supercurrents without macroscopic magnetic stray fields, opening new opportunities for superconducting spintronic applications.

cond-mat.supr-con↗
arXiv2026

A linear bound for a connectivity partition in graphs

Kühn and Osthus proved that, for every positive integer $\ell$, every $2^{16}\ell^2$-connected graph $G$ has a partition $V(G)=S\cup T$ such that $G[S]$ and $G[T]$ are $\ell$-connected and $d_T(v)\geq \ell$ for every $v\in S$. And they asked whether the quadratic bound can be replaced by a linear bound. In this paper, we answer this question in the affirmative by proving that every $641\ell$-connected graph $G$ has a partition $V(G)=S\cup T$ such that $G[S]$ and $G[T]$ are $\ell$-connected and $d_T(v)\ge \ell$ for every $v\in S$. The proof integrates the Moser-Tardos resampling algorithm.

math.CO↗
arXiv2026

SL-RFSIM: Enabling Scalable Multi-Hop 5G NR Sidelink Mesh Networking in OpenAirInterface

Recent 3GPP releases have extended 5G New Radio (NR) Sidelink (SL) to support device-to-device (D2D) relay and multi-hop capabilities. This paper presents SL-RFSIM, a component-based experimentation framework extending OpenAirInterface (OAI) with scalable multi-hop NR SL capabilities. SL-RFSIM replaces the legacy OAI RF simulator with a broker-based publish/subscribe architecture enabling arbitrary peer-to-peer connectivity while preserving compatibility with the OAI protocol stack. The framework further integrates pluggable mobility, propagation, reception, and monitoring services, and supports Layer-2 mesh networking through BATMAN-adv. Experimental evaluation on the SLICES-RI research infrastructure validates the proposed architecture through representative mesh networking scenarios and identifies the current software bottlenecks limiting scalability. SL-RFSIM provides an open-source foundation for reproducible experimental research on 5G NR SL and future multi-hop cellular mesh networks.

cs.NI↗
arXiv2026

ThinkingGuard: Decoding Implicit Hazards via Step-by-Step Risk Attribution in Multimodal Large Language Models

While Multimodal Large Language Models (MLLMs) are increasingly deployed in safety-critical domains, their reliability is threatened by multimodal implicit risks. Unlike explicit threats, these hazards emerge when individually benign text and neutral visual entities logically converge to induce unsafe outputs. Current detection methods fail to address this because they overlook the underlying risk activation mechanisms that govern cross-modal risk activation, leading to single-modality shortcut learning and hallucinated rationalizations. To bridge this gap, we first construct TriggerBench, the first dataset explicitly modeling risk compositionality (5,600 instances). By formally isolating Key Elements and Trigger Elements to build counterfactual contrastive pairs, TriggerBench eliminates risk residues and forces models to perform genuine logical deduction rather than superficial pattern matching, which provides a rigorous foundation for both large-scale training and fine-grained evaluation. Building on this, we propose a Step-Supervised Structured Reasoning training framework and employ it to train ThinkingGuard, a specialized guard model. Inspired by Situation Awareness theory, we decouple implicit risk identification into progressive cognitive stages, and utilize a step-reward Monte Carlo Tree Search algorithm to explore optimal reasoning trajectories, which are then distilled into the model through Dual-Constraint Preference Alignment. Extensive experiments across both standard and implicit safety benchmarks demonstrate that ThinkingGuard achieves strong performance. Project resources are available at https://github.com/FroggyChen/ThinkingGuard.

cs.CV↗
arXiv2026

The $m$-Laplace equation with a gradient term II: classification in the second critical case

Let $1<m<n$ and $0<q<m-1$. We classify positive, locally bounded weak solutions of \[-Δ_m u=u^p|Du|^q\quad\text{in }\mathbb R^n,\qquad p=\frac{m-q}{n-m}\left(n+\frac{q}{m-1-q}\right)-1.\] Every solution is constant or belongs to an explicit family of radial solutions, up to translation and scaling. No global bound, decay, or finite-energy assumption is required. The proof is based on a scalar differential identity and a maximum principle that remains valid at critical points. These lead to a sharp gradient bound, classification of normalized solutions, and a first-contact argument for arbitrary entire solutions. Local regularity and the behavior at critical points are treated throughout in the weak solution class; global $C^2$ regularity is not assumed.

math.AP↗
arXiv2026

Sharp connectivity thresholds for mixed rigidity packings and improved bounds for highly connected orientations of graphs

Garamvölgyi, Jordán, Király and Villányi [{{\bf Forum Math. Pi} \textbf{13} (2025), Paper No.~e11}] posed two sharp connectivity conjectures for packing rigid spanning subgraphs: one for the equal-dimensional case and the other for the packing of a $d$-rigid spanning subgraph with a spanning tree. We prove a unified theorem: for arbitrary positive integers $d_1,\ldots,d_s$, every $\sum_{i=1}^{s}d_i(d_i+1)$-connected graph contains pairwise edge-disjoint spanning subgraphs $H_1,\ldots,H_s$ such that $H_i$ is $d_i$-rigid for every $i$. The connectivity bound is sharp whenever $\sum_{i=1}^{s}d_i(d_i+1)\ge4$. As special cases, the theorem settles both conjectures, confirms the conjecture of Garamvölgyi, Jordán and Király [{\bf J. Combin. Theory Ser. B} \textbf{166} (2024), 1--29] that every $tk(k+1)$-connected graph contains $t$ pairwise edge-disjoint $k$-connected spanning subgraphs, and gives the sharp threshold $d(d+1)+2r$ for packing one $d$-rigid spanning subgraph together with $r$ pairwise edge-disjoint spanning trees. We also obtain two upper bounds related to Thomassen's conjecture on highly connected orientations of graphs. If $f(q)$ is the least integer such that every $f(q)$-connected graph has a $q$-connected orientation, then $f(q)\le(25q^2+41q-16)/2$ for every $q\ge3$ and $f(q)\le8q^2+212q+1404=(8+o(1))q^2$ for all sufficiently large $q$; these two results reduce the leading coefficient in the previous quadratic bound from $320$ to $25/2$ for every $q\ge3$ and $8$ for all sufficiently large $q$. Compared to the bound for $f(q)$ obtained by Garamvölgyi et al. we obtain the better bounds, not only through our tight rigidity result but also by exploiting the leftover edges when we remove two edge-disjoint spanning (sufficiently) rigid graphs.

math.CO↗
arXiv2026

Interior estimates for Hessian quotient equations

We prove interior Hessian estimates in arbitrary dimensions for admissible solutions of Hessian quotient equations with constant right-hand side, when the numerator and denominator differ in degree by one or two. Counterexamples show that this degree range is optimal. The proof combines a new nonlinear comparison function, a doubling maximum principle, and a Pogorelov estimate.

math.AP↗