arXiv subjects

Shuo Wei

Publications and source records attributed to Shuo Wei.

13 recordsLinked to original sources

arXiv2026

Cycles in Regular Multipartite Tournaments

We study balanced cycles in regular multipartite tournaments. A cycle in a $c$-partite tournament is $k$-balanced if it contains exactly $k$ vertices from each partite class. We prove that there exists an absolute integer $r_0$ such that every regular $c$-partite tournament with each partite class size of $r\ge r_0$ contains a $k$-balanced cycle for every $1\le k\le r$ when $c=3$ or $c\ge5$, and for every $2\le k\le r$ when $c=4$. In particular, our result confirms three conjectures of Volkmann and Winzen for large regular multipartite tournaments. We also confirm Yeo's conjecture on the existence of $(p,q)$-cycle factors for large regular $c$-partite tournaments $G$ with $c\ge4$, where a $(p,q)$-cycle factor is a pair of vertex-disjoint cycles of lengths $p$ and $q$ spanning $G$.

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

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

Degree conditions for $k$-strong orientations of digraphs

Jackson and Thomassen conjectured that every $2k$-strong digraph contains a spanning $k$-strong oriented subdigraph. We prove sharp degree conditions for the existence of such a subdigraph. For every fixed positive integer $k$ and all sufficiently large $n$, every $n$-vertex digraph $D$ with $δ^0(D)\ge\lfloor(n+k-1)/2\rfloor$ admits a $k$-strong orientation. This threshold is best possible even for the weaker conclusion that $D$ itself is $k$-strong. We also prove a sharp Woodall-type analogue for every fixed positive integer $k$ and all sufficiently large $n$: if $d_D^+(x)+d_D^-(y)\ge n+2k-2$ for every missing arc $xy$, then $D$ admits a $k$-strong orientation, and this bound is again best possible. As a further consequence, we determine the sharp minimum total degree threshold. Finally, the semi-degree result also remains valid when $k\leαn$ for every fixed $0<α<0.094882\ldots$.

math.CO↗

arXiv2026

An Exact Dominant Degree Condition for Transitive Tournament Factors in Digraphs

Let $r\ge2$, let $T_r$ denote the transitive tournament on $r$ vertices, and write $d_G^*(v):=\max\{d_G^+(v),d_G^-(v)\}$. We prove that if $r\mid n$ and an $n$-vertex digraph $G$ satisfies $d_G^*(x)+d_G^*(y)\ge 2(1-1/r)n-1$ for every $x\ne y \in V(G)$ with $xy \notin E(G)$, then $G$ has a $T_r$-factor, and the bound is best possible. Furthermore, by applying our main theorem, we settle Treglown's conjecture on the dominant degree $d^*_G(x) \ge (1-1/r)n$ and answer Molla and Treglown's problem of determining the exact Ore-type threshold $2(1-1/r)n - 1$, and we obtain stronger versions of the theorems of Czygrinow, DeBiasio, Kierstead and Molla.

math.CO↗

arXiv2026

Emergent Equilibrium Structure Along a Critical Cluster Recursion

Critical universality does not determine the microscopic conditional structure of a probability measure. We study a bicolored cluster recursion constrained to remain critical at every generation, with no equilibrium spin measure or fixed coupling imposed. In both two and three dimensions, the resulting history-dependent sequence develops a common Ising/Fortuin--Kasteleyn (FK) compatibility structure: the second-shell dependence of a one-site conditional law is strongly suppressed, nearest-neighbor effective couplings move progressively toward one another near the critical Ising value, and cluster and interface observables organize around the corresponding FK geometry. An exact cluster-coloring factorization singles out $q=2$ as the point where the residual connectivity weight disappears from the two-color spin marginal. Thus equilibrium-compatible conditional structure can emerge along a trajectory that remains critical throughout.

cond-mat.stat-mech↗

arXiv2026

Spanning $H$-subdivisions with Prescribed Path Lengths

We study spanning $H$-subdivisions in dense graphs where the length of every subdivision path is prescribed in advance. This problem is motivated in part by a question of Pavez-Signé [Combin. Probab. Comput. 33 (2024), 121--128], who asked whether the subdivision paths in a spanning $H$-subdivision can be required to have similar lengths. Let $h\ge3$ be an integer and let $0<β\llα\ll1/h$. We prove that, for all sufficiently large $n$, every $n$-vertex graph $G$ with $δ(G)\ge n/2+\lfloor h/3\rfloor$ has the following property. For every graph $H$ with $h$ edges and no isolated vertices, write $E(H)=\{e_1,\ldots,e_h\}$, and every choice of integers $\ell_1,\ldots,\ell_h\ge4$ satisfying $\sum_{i=1}^h\ell_i=n-|V(H)|+h$ and $\sum_{\ell_i<αn}\ell_i\leβn$, the graph $G$ contains a spanning $H$-subdivision in which the $i$th edge of $H$ is replaced by a path of length exactly $\ell_i$. We also give a family of examples showing that a linear additive term in $h$ is necessary in general.

math.CO↗

arXiv2026

Nearly balanced spanning subdivisions in dense digraphs

Pavez-Signé [Combin. Probab. Comput. 33 (2024), 121--128] conjectured a Dirac-type condition for spanning $H$-subdivisions and asked whether the subdivision paths can additionally be required to have similar lengths. Lee [European J. Combin. 124 (2025), 104059] resolved the existence conjecture in the stronger setting of digraphs. We answer the length-control question in this stronger directed setting: for every $\varepsilon>0$, there exists a constant $C_0>0$ such that, for every digraph $H$ with $h$ arcs and no isolated vertices, every $n$-vertex digraph $D$ with $n\ge C_0h$ and $δ^0(D)\ge(1/2+\varepsilon)n$ contains a spanning $H$-subdivision whose subdivision paths have lengths differing by at most one.

math.CO↗

arXiv2026

The exact total degree threshold for the square of a Hamilton cycle in digraphs

The Pósa-Seymour conjecture establishes the minimum degree threshold required to guarantee the presence of the $k$th power of a Hamilton cycle in a graph. Following numerous partial results, Komlós, Sárközy, and Szemerédi confirmed the conjecture holds for all sufficiently large graphs. Treglown later conjectured the analogous minimum semi-degree threshold for forcing the $k$th power of a Hamilton cycle in a digraph. Subsequently, DeBiasio et al. proposed a conjecture on the minimum total degree threshold for the same problem. In this paper we settle the conjecture of DeBiasio et al. for $k=2$. Specifically, we prove that every sufficiently large $n$-vertex digraph with minimum total degree at least $8n/5-c$ contains the square of a Hamilton cycle, where $c=2$ if $n\equiv2,4\pmod 5$, and $c=1$ otherwise.

math.CO↗

arXiv2026

Universality of percolation at dynamic pseudocritical point

Universality, encompassing critical exponents, scaling functions, and dimensionless quantities, is fundamental to phase transition theory. In finite systems, universal behaviors are also expected to emerge at the pseudocritical point. Focusing on two-dimensional percolation, we show that the size distribution of the largest cluster asymptotically approaches to a Gumbel form in the subcritical phase, a Gaussian form in the supercritical phase, and transitions within the critical finite-size scaling window. Numerical results indicate that, at consistently defined pseudocritical points, this distribution exhibits a universal form across various lattices and percolation models (bond or site), within error bars, yet differs from the distribution at the critical point. The critical polynomial, universally zero for two-dimensional percolation at the critical point, becomes nonzero at pseudocritical points. Nevertheless, numerical evidence suggests that the critical polynomial, along with other dimensionless quantities such as wrapping probabilities and Binder cumulants, assumes fixed values at the pseudocritical point that are independent of the percolation type (bond or site) but vary with lattice structures. These findings imply that while strict universality breaks down at the pseudocritical point, certain extreme-value statistics and dimensionless quantities exhibit quasi-universality, revealing a subtle connection between scaling behaviors at critical and pseudocritical points.

cond-mat.stat-mech↗

arXiv2026

Oriented Discrepancy of The Square of Hamilton Cycles

For an oriented graph $G$, the oriented discrepancy problem concerns the existence of a spanning subgraph of $G$ with a large imbalance between its forward and backward edge orientations. Freschi and Lo proved the Dirac-type Hamilton cycle result in oriented graphs, and asked for an analogue for powers of Hamilton cycles under a minimum-degree condition. We show that, for sufficiently large $n$, every oriented graph $G$ on $n$ vertices with minimum degree $δ(G)\geq 2n/3$ contains the square of a Hamilton cycle $H$ with $σ_{\max}(H)$ guaranteed to exceed a function depending on $δ(G)$ and $n$.

math.CO↗

arXiv2026

Evolving fractal dimensions in iterative bicolored percolation

Criticality is traditionally regarded as an unstable, fine-tuned fixed point of the renormalization group. We introduce an iterative bicolored percolation process in two dimensions and show that it can both preserve criticality and transform fractal dimensions. Starting from critical configurations, such as the O$(n)$ loop and fuzzy Potts models, successive coarse-graining generates a hierarchy of distinct yet critical generations. Using the conformal loop ensemble, we derive exact, generation-dependent fractal dimensions, which are quantitatively confirmed by large-scale Monte Carlo simulations. The evolutionary trajectory depends not only on the universality class of the initial state but also on whether it possesses a two-state critical structure, leading to different critical exponents starting from site and bond percolation. These results establish a general geometric mechanism for evolving fractal dimensions, in which scale invariance persists across generations.

cond-mat.stat-mech↗