arXiv subjects

Zhilan Wang

Publications and source records attributed to Zhilan Wang.

14 recordsLinked to original sources

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-Sign\'e [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 $0<\beta\ll\alpha\ll1/h$. We prove that, for all sufficiently large $n$, every $n$-vertex graph $G$ with $\delta(G)\ge n/2+\lfloor h/3\rfloor$ 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 $\sum_{\ell_i<\alpha n}\ell_i\le\beta n$, 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.

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arXiv2026

Nearly balanced spanning subdivisions in dense digraphs

Pavez-Sign\'e [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 $\delta^0(D)\ge(1/2+\varepsilon)n$ 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 P\'{o}sa-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, Koml\'{o}s, S\'{a}rk\"{o}zy, and Szemer\'{e}di 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

Ramsey-Tur\'an Type Problem for Perfect Transitive Triangle Tilings in Digraphs

The classical Corr\'adi-Hajnal theorem states that for any multiple $n$ of $3$, if $G$ is a graph with $n$ vertices and $\delta(G) \geq 2n/3$, then $G$ can be partitioned into $n/3$ vertex-disjoint copies of the triangle [\emph{Acta Math. Acad. Sci. Hung.}, 14:423-439, 1964]. Balogh, Molla and Sharifzadeh obtained a smaller lower bound by adding the independence number condition [\emph{Random Struct. Algorithms}, 49:669-693, 2016]. In this paper, we study perfect tilings in digraphs subject to conditions on the independence number and the degree. The independence number, $\alpha(D)$, of $D$ is the maximum integer $k$ such that $D$ has an independent set of cardinality $k$. We show that if $D$ is an $n$-vertex digraph with $\alpha(D)\leq o(1)n$ and $\delta(D) \geq (1+o(1))n$, then $D$ has a perfect $T_3$-tiling, where $T_3$ denotes a transitive triangle. This minimum degree condition is asymptotically best possible. Moreover, our result implies the theorem of Balogh, Molla, and Sharifzadeh concerning perfect triangle tilings.

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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 $\delta(G)\geq 2n/3$ contains the square of a Hamilton cycle $H$ with $\sigma_{\max}(H)$ guaranteed to exceed a function depending on $\delta(G)$ and $n$.

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arXiv2026

The $H$-linkage problems in sparse robustly expanding digraphs

The Nash-Williams conjecture establishes degree sequence conditions ensuring Hamilton cycles in digraphs. An asymptotic version of this conjecture for large digraphs was independently derived by several researchers. We strengthen these results by proving the following results under the same asymptotic degree sequence conditions. For any digraph $H$, a digraph $D$ is $(\mathcal{N}H)$-linked if there exists an integer $l_0$ such that for any vertex set $U$ of cardinality $|V(H)|$ and every integer set $\mathcal{N}=\{l_i\}_{i=1}^{|A(H)|}$ with $l_i\geq l_0$, $D$ contains an $H$-subdivision with $U$ as branch-vertex set and the values in $\mathcal{N}$ specifying the lengths of the subdivided paths. Let $D$ be a sufficiently large digraph of order $n$ with the out-degree sequence $d_1^+\leq\cdots\leq d_n^+$ and the in-degree sequence $d_1^-\leq\cdots\leq d_n^-$. We prove that if for every $\gamma\in(0, 1)$ and every integer $0\leq i<n/2$, the following conditions hold: (i) $d_i^+\geq i+\gamma n$ or $d_{n-i-\gamma n}^-\geq n-i$, and (ii) $d_i^-\geq i+\gamma n$ or $d_{n-i-\gamma n}^+\geq n-i$, then $D$ is $(\mathcal{N}H)$-linked, and also admits a perfect $H$-subdivision tiling with subdivision orders $\{n_1, \ldots, n_k\}$, where each $n_i\geq C_0$ for some integer $C_0$.

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arXiv2026

An Ore-type Theorem for Oriented Discrepancy of Hamilton Cycles

Oriented graph discrepancy problems focus on finding specific subgraphs within a given oriented graph $G$ that contain a significant number of edges in one direction. This concept was first introduced by Gishboliner, Krivelevich, and Michaeli, and has since been further investigated by Freschi and Lo [J. Combin. Theory, Ser. B 169 (2024)], who gave a tight lower bound for the discrepancy of Hamilton cycles in terms of the minimum degree of $G$. Furthermore, they raised the problem of extending such results to Ore-type conditions. Here, an Ore-type condition refers to the minimum degree-sum of non-adjacent vertices, formally defined as: $\sigma_2(G)=\min\{d(x)+d(y)\mid x, y \in V(G) \text{ and } xy \notin E(G)\}$. In this paper, we address this question by showing that for every sufficiently large oriented graph $G$, if $\sigma_2(G)\geq n$, then $G$ contains a Hamilton cycle $C$ with at least $\max\{n/2,\sigma_2(G)/2-o(n)\}$ edges in one direction. Moreover, this result is asymptotically tight.

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arXiv2024

Spanning $H$-subdivisions and perfect $H$-subdivision tilings in dense digraphs

Given a (di)graph $H$, we say that a (di)graph $H^\prime$ is an $H$-subdivision if $H^\prime$ is obtained from $H$ by replacing one or more edges with internally vertex-disjoint path(s). Pavez-Sign\'{e} conjectured that for every $\varepsilon>0$, there exists a constant $C_0>0$ such that for every graph $H$ with $h$ edges and no isolated vertices, if $G$ is a graph on $n\geq C_0h$ vertices and minimum degree $\delta(G)\geq(1+\varepsilon)\frac{n}{2}$, then $G$ contains a spanning $H$-subdivision. This conjecture was later resolved by Lee [European J. Combin. \textbf{124} (2025), 104059]. In this paper, we strengthen Lee's result. Specifically, we prove that for any digraph $D$ on $n\geq C_0h$ vertices, if the minimum semi-degree of $D$ is at least $\frac{n+h}{2}-1$, then $D$ contains a spanning $H$-subdivision. The lower bound on the minimum semi-degree is best possible. Furthermore, we show that there exist constants $C>0$ and $\alpha, \beta\in(0, 1)$ such that for any integer partition $n=n_1+\cdots+n_m\geq Cm$ with $n_i\geq|V(H)|+3h$ for each $i$, and $\sum_{n_i<\alpha n}n_i\leq\beta n$, if a digraph of order $n\geq Cm$ has minimum semi-degree at least $\frac{n+m+h}{2}-1$, then it contains $m$ vertex-disjoint $H$-subdivisions whose orders are $n_1, \ldots, n_m$, respectively. The bound $\frac{n+m+h}{2}-1$ is also optimal. This work partly answers a conjecture of Lee [Combin. Probab. Comput. \textbf{34} (2025), 421--444] and generalizes a recent result from the same paper.

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arXiv2024

A generalization of the Hamiltonian cycle in dense digraphs

Let D be a digraph and C be a cycle in D. For any two vertices x and y in D, the distance from x to y is the minimum length of a path from x to y. We denote the square of Let $D$ be a digraph and $C$ be a cycle in $D$. For any two vertices $x$ and $y$ in $D$, the distance from $x$ to $y$ is the minimum length of a path from $x$ to $y$. We denote the square of the cycle $C$ to be the graph whose vertex set is $V(C)$ and for distinct vertices $x$ and $y$ in $C$, there is an arc from $x$ to $y$ if and only if the distance from $x$ to $y$ in $C$ is at most $2$. The reverse square of the cycle $C$ is the digraph with the same vertex set as $C$, and the arc set $A(C)\cup \{yx: \mbox{the vertices}\ x, y\in V(C)\ \mbox{and the distance from $x$ to $y$ on $C$ is $2$}\}$. In this paper, we show that for any real number $γ>0$ there exists a constant $n_0=n_0(γ)$, such that every digraph on $n\geq n_0$ vertices with the minimum in- and out-degree at least $(2/3+γ)n$ contains the reverse square of a Hamiltonian cycle. Our result extends a result of Czygrinow, Kierstead and Molla.

math.CO↗

arXiv2024

Cycle-factors in oriented graphs

Let $k$ be a positive integer. A $k$-cycle-factor of an oriented graph is a set of disjoint cycles of length $k$ that covers all vertices of the graph. In this paper, we prove that there exists a positive constant $c$ such that for $n$ sufficiently large, any oriented graph on $n$ vertices with both minimum out-degree and minimum in-degree at least $(1/2-c)n$ contains a $k$-cycle-factor for any $k\geq4$. Additionally, under the same hypotheses, we also show that for any sequence $n_1, \ldots, n_t$ with $\sum^t_{i=1}n_i=n$ and the number of the $n_i$ equal to $3$ is $αn$, where $α$ is any real number with $0<α<1/3$, the oriented graph contains $t$ disjoint cycles of lengths $n_1, \ldots, n_t$. This conclusion is the best possible in some sense and refines a result of Keevash and Sudakov.

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arXiv2024

The generalizations of Hamiltonian in oriented graphs

An oriented graph is an orientation of a simple graph. In 2009, Keevash, Kühn and Osthus proved that every sufficiently large oriented graph $D$ of order $n$ with $(3n-4)/8$ is Hamiltonian. Later, Kelly, Kühn and Osthus showed that it is also pancyclic. Inspired by this, we show that for any given constant $t$ and positive integer partition $n = n_1 + \cdots + n_t$, if $D$ is an oriented graph on $n$ vertices with minimum semidegree at least $(3n-4)/8$, then it contains $t$ disjoint cycles of lengths $n_1,\ldots , n_t$. Also, we determine the bounds on the semidegree of sufficiently large oriented graphs that are strongly Hamiltonian-connected, $k$-ordered Hamiltonian and spanning $k$-linked.

math.CO↗

arXiv2024

A Dirac-type theorem for arbitrary Hamiltonian $H$-linked digraphs

Given any digraph $D$ on $n$ vertices, let $\mathcal{P}(D)$ be the family of all directed paths in $D$, and let $H$ be a digraph with the arc set $A(H)=\{a_1, \ldots, a_k\}$. The digraph $D$ is called arbitrary Hamiltonian $H$-linked if for any injective map $f: V(H)\rightarrow V(D)$ and any integer set $\mathcal{N}=\{n_1, \ldots, n_k\}$ satisfying that $n_i\geq4$ for each $i\in\{1, \ldots, k\}$, there is a map $g: A(H)\rightarrow \mathcal{P}(D)$ such that for every arc $a_i=uv$, $g(a_i)$ is a directed path from $f(u)$ to $f(v)$ of length $n_i$, and different arcs are mapped into internally vertex-disjoint directed paths in $D$, and $\bigcup_{i\in[k]}V(g(a_i))=V(D)$. Here, the length of a directed path is defined as the number of its arcs. In this paper, we prove that for any digraph $H$ with $k$ arcs and $\delta(H)\geq1$, there exists a constant $C_0=C_0(k)$ such that if $D$ is a digraph of order $n\geq C_0$ and minimum in- and out-degree at least $n/2+k$, then it is arbitrary Hamiltonian $H$-linked. The lower bound on the minimum in- and out-degree is best possible. We further prove a more general form that allows $k$ to be linear in $n$, while imposing some restrictions on the lengths of the subdivided arcs. As corollaries, we solved a conjecture of Wang \cite{Wang} for sufficiently large graphs, and partly answered a problem raised by Pavez-Sign\'{e} \cite{Pavez}.

math.CO↗

arXiv2023

A characterization of rich c-partite (c > 7) tournaments without (c + 2)-cycles

Let c be an integer. A c-partite tournament is an orientation of a complete c-partite graph. A c-partite tournament is rich if it is strong, and each partite set has at least two vertices. In 1996, Guo and Volkmann characterized the structure of all rich c-partite tournaments without (c + 1)-cycles, which solved a problem by Bondy. They also put forward a problem that what the structure of rich c-partite tournaments without (c + k)-cycles for some k>1 is. In this paper, we answer the question of Guo and Volkmann for k = 2.

math.CO↗

arXiv2016

Tautological Integrals on Hilbert Schemes of Points on Curves

We propose a conjecture on the generating series of Chern numbers of tautological bundles on Hilbert schemes of points on curves and establish the rank 1 and rank -1 case of this conjecture. Thus we compute explicitly the generating series of integrals of the Segre classes of tautological bundles of line bundles on curves, which has a similar structure as Lehn's conjecture for surfaces.

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