arXiv subjects

Jiangdong Ai

Publications and source records attributed to Jiangdong Ai.

At least 19 recordsLinked to original sources

arXiv2026

A linear bound for nested cycles without geometric crossings

Cycles $C_1,\ldots,C_k$ in a graph are called nested without geometric crossings if they are pairwise edge-disjoint, $V(C_k)\subseteq\cdots\subseteq V(C_1)$, and each pair of consecutive cycles induces the same cyclic order on the vertices of the inner cycle, up to reversal. Let $f_k(n)$ be the least number of edges that forces such a family in every $n$-vertex graph. Gil Fernández, Kim, Kim and Liu proved that $f_2(n)=O(n)$, answering a question of Erdős, and asked whether $f_k(n)=O_k(n)$ for every fixed $k$. We prove this for all $k$. The proof selects the inner cycles together with a disjoint subgraph that supplies their external neighbours. A reselection argument gives disjoint paths from every inner-cycle vertex to any sufficiently large target set. Sublinear expansion and a rooted clique minor then allow the vertices to be joined in the required cyclic order.

math.CO↗

arXiv2026

An improved finite bound for oriented trees in tournaments

Sumner's universal tournament conjecture asserts that every tournament on $2n-2$ vertices contains every oriented tree on $n$ vertices. Let $f(n)$ be the least integer $N$ such that every tournament on $N$ vertices contains every oriented tree on $n$ vertices. Havet and Thomassé proved that $f(n)\le \lceil(7n-5)/2\rceil$, El Sahili improved this to $f(n)\le3n-3$, and Dross and Havet subsequently obtained $f(n)\le\lceil21n/8-47/16\rceil$. We refine their median-order method. More precisely, every non-bi-arborescence on $n$ vertices with $k$ leaves is $(4n-2k-4)$-unavoidable, which strictly improves their many-leaf estimate; bi-arborescences satisfy the stronger bound $2n-2$. Combining this refinement with their few-leaf bound gives $f(n)\le\lceil(18n-23)/7\rceil$ for every $n\ge2$. Thus the coefficient in the previously best general bound valid uniformly for all $n$ is reduced from $21/8$ to $18/7$.

math.CO↗

arXiv2026

Improved Bounds for Unavoidable Claws in Tournaments

Let $u(n)$ be the largest integer $d$ such that every $n$-vertex claw with at most $d$ branches occurs in every tournament on $n$ vertices, and let $c_{\mathrm{claw}}=\limsup_{n\to\infty}u(n)/n$. In 1998, Lu, Wang and Wong proved that $19/50\le c_{\mathrm{claw}}\le11/23$, and these have remained the best bounds known. We improve them to $2/5\le c_{\mathrm{claw}}\le10/21$. We also isolate two parameters $θ$ and $σ$ which place the lower- and upper-bound arguments in a common framework: we show $σ\leθ$ and $1/21\leσ\leθ\le1/5$, our two bounds being the images of the endpoints under $α\mapsto\frac12-\fracα{2}$, and $σ=θ$ would force $\lim u(n)/n$ to exist.

math.CO↗

arXiv2026

The Complexity of Mixed Arc-Disjoint Spanning Subdigraphs with Antistrong Connectivity

A trail is antidirected if its arcs alternate between forward and backward. A digraph $D$ is antistrong if, for every ordered pair of distinct vertices $x,y\in V(D)$, it contains a forward antidirected $(x,y)$-trail. Bang-Jensen, Bessy, Jackson and Kriesell [J. Combin. Theory Ser. B 122 (2017), 68--90] introduced antistrong connectivity and posed two problems concerning mixed arc-disjoint spanning subdigraphs. In the first problem, one seeks an antistrong spanning subdigraph and an arc-disjoint strong spanning subdigraph. In the second, strong connectivity is replaced by the requirement that the underlying graph of the second subdigraph be 2-edge-connected. Bang-Jensen et al. asked whether each of the two problems can be solved in polynomial time. We prove that the two associated decision problems are NP-complete. The first remains NP-complete for digraphs with maximum out-degree at most four and maximum in-degree at most five. The second remains NP-complete even for oriented digraphs that are strong and antistrong, whose underlying graphs are 3-vertex-connected, and in which all but at most two vertices have both in-degree and out-degree at most four. In particular, the latter hardness result does not rely on digons.

cs.DM↗

arXiv2026

A Linear Lower Bound for Dominating Sets in $k$-Majority Tournaments

A $k$-majority tournament on a finite vertex set is defined by $2k-1$ linear orders, with $u\to v$ when $u$ lies above $v$ in at least $k$ of the orders. Let $F(k)$ be the maximum, over all $k$-majority tournaments, of the size of a minimum dominating set. Alon, Brightwell, Kierstead, Kostochka, and Winkler proved that $C_1k/\log k \leq F(k) \leq C_2k\log k$ for suitable positive constants $C_1$ and $C_2$. In this paper, we prove the linear lower bound $F(k)\ge \left\lfloor\frac{k+1}{2}\right\rfloor $ for $k\ge 3$.

math.CO↗

arXiv2026

On Seymour's and Sullivan's Second Neighbourhood Conjectures

For a vertex $x$ of a digraph, $d^+(x)$ ($d^-(x)$, resp.) is the number of vertices at distance 1 from (to, resp.) $x$ and $d^{++}(x)$ is the number of vertices at distance 2 from $x$. In 1995, Seymour conjectured that for any oriented graph $D$ there exists a vertex $x$ such that $d^+(x)\leq d^{++}(x)$. In 2006, Sullivan conjectured that there exists a vertex $x$ in $D$ such that $d^-(x)\leq d^{++}(x)$. We give a sufficient condition in terms of the number of transitive triangles for an oriented graph to satisfy Sullivan's conjecture. In particular, this implies that Sullivan's conjecture holds for all orientations of planar graphs and of triangle-free graphs. An oriented graph $D$ is an oriented split graph if the vertices of $D$ can be partitioned into vertex sets $X$ and $Y$ such that $X$ is an independent set and $Y$ induces a tournament. We also show that the two conjectures hold for some families of oriented split graphs, in particular, when $Y$ induces a regular or an almost regular tournament.

math.CO↗

arXiv2026

Finite palette endpoints and degree-square Turán problems

We study finite palette extremal problems motivated by uniform Turán densities of $3$-uniform hypergraphs. Given a self-converse tournament $T$ with at least two vertices, we determine the largest possible number of admissible triples in an $m$-color palette that avoids the left and right palettes associated with $T$. The answer is the one-sided degree-square Turán number \[ \operatorname{pal}_T(m) = \operatorname{ex}_2^+(m,T) = \max\left\{ \sum_{v\in V(D)} d_D^+(v)^2: |V(D)|=m,\ D\text{ is }T\text{-free} \right\}. \] Thus this palette problem is reduced to an extremal problem for digraph out-degrees. We then prove a prefix-majorization lemma for convex out-degree moments and apply it to directed cycles. In particular, $\operatorname{ex}_2^+(m,\overrightarrow C_3)=\frac{m(m^2-1)}3$, which gives the sharp $m$-color palette endpoint $\frac13-\frac{1}{3m^2}$ for the directed triangle. Combining this endpoint with the palette characterization of uniform Turán density and the palette separation theorem, we show that for every $m\ge2$ there is a finite $3$-graph $H_m$ such that \[ \frac13-\frac{1}{3m^2}\le π_u(H_m)\le \frac13. \] Hence there is a sequence of individual finite $3$-graphs whose uniform Turán densities converge to $1/3$. We also describe the extremal palettes, prove a qualitative edit-distance stability theorem, and compute the Lagrangian of the endpoint palette $\mathcal P_m^\star$. As a consequence, for every $m\ge2$ there is a finite family $\mathcal F_m$ of $3$-graphs with $π_u(\mathcal F_m)=\frac13-\frac{1}{3m^2}$, so $1/3$ is an accumulation point of uniform Turán densities of finite forbidden families.

math.CO↗

arXiv2026

Triangular cutoff threshold for the inversion walk on tournaments and the state space of restricted inversions

Given a labelled tournament on $[n]$, \emph{inverting} a vertex subset $X$ means reversing every edge with both endpoints in $X$. Alon, Powierski, Savery, Scott, and Wilmer~\cite{AlonPowierskiSaveryScottWilmer2024} asked for the mixing time of the Markov chain that repeatedly inverts a uniformly random subset of $[n]$. We show that this \emph{inversion walk} has a triangular cutoff threshold. Let $T(s):=\binom{s+1}{2}$. For every integer sequence $s_n\in[0,n]$, $$ d_n(n-s_n)\longrightarrow \begin{cases} 1,& T(s_n)-n\to+\infty, 0,& T(s_n)-n\to-\infty. \end{cases} $$ Consequently, for every fixed $\varepsilon\in(0,1)$, $ t_{\mix}^{(n)}(\varepsilon)=n-\sqrt{2n}+O(1).$ We also prove quantitative one-sided bounds, with absolute constants $C_\star<0.36$ and $κ<3.47$, $$ d_n(n+c)\le C_\star 2^{-c}\quad(c\ge0), \qquad d_n(n-s)\ge 1-κ\,2^{\,n-\binom{s+1}{2}}\quad(0\le s\le n). $$ As a second result, we characterise the state space of the \emph{$k$-restricted inversion walk}, which inverts a uniformly random $k$-subset at each step. For $n\ge 4$ and $2\le k\le n-2$, the reachable states form a coset of a subgroup $H_k\le\F_2^{\binom{n}{2}}$ whose defining parity constraints are determined by $k\bmod 4$; equivalently, its codimension is $0,1,n-1$, or $n$ according as $k\equiv2,0,3$, or $1\pmod4$.

math.CO↗

arXiv2026

Tree suspensions and transfer functions for single degree Turán spectra

For integers $1\le \ell<k$, let $Π^k_\ell$ denote the single-forbidden $\ell$-degree Turán spectrum of $k$-uniform hypergraphs. We introduce transfer functions for this spectrum: explicit functions $f$ such that, for every $F$, there is another single $k$-graph $F^*$ with $π_\ell(F^*)=f(π_\ell(F))$. This gives a mechanism for producing new single-forbidden densities while retaining full control of the resulting value. Our transfer functions are realized by a new family of suspension-type operations, called tree suspensions. From these operations we obtain three explicit maps: one acting on $Π^k_\ell$ for every $1\le\ell<k$, a second acting when $\ell\ge k/2$, and a third acting in the ordinary Turán case $\ell=1$. The common feature is a robust tree structure which gives the lower bound by a two-part construction and, in the regimes above, admits a matching embedding or Lagrangian upper bound. As a first application, the universal transfer function propagates accumulation points. Using the recent zero-accumulation results for $\ell\ge2$ together with the ordinary Turán accumulation result of Conlon and Schülke, we prove that $Π^k_\ell$ has infinitely many accumulation points for every $k\ge3$ and every $1\le\ell<k$. This recovers, in particular, the known infinitude of accumulation points in the ordinary and codegree spectra. As a second application, combining two independent transfer functions forces algebraic degrees to grow. For every $k\ge3$ and every $\ell\in\{1,\lceil k/2\rceil,\ldots,k-2\}$, the spectrum $Π^k_\ell$ contains algebraic numbers of arbitrarily large degree over $\mathbb Q$. Thus the arithmetic complexity previously known for finite forbidden families already occurs in the single-forbidden spectrum, both for ordinary Turán density and for a broad range of degree Turán densities.

math.CO↗

arXiv2026

Vanishing orders, suspensions and zero degree Turán densities

For integers $1\le \ell<k$, the $\ell$-degree Turán density $π_\ell(F)$ measures the minimum $\ell$-degree threshold that forces a copy of a fixed $k$-uniform hypergraph $F$, generalizing both the classical Turán density $π_1$ and the codegree Turán density $π_{k-1}$. Motivated by Erdős' characterization of $k$-graphs with zero Turán density, we study the structural implications of vanishing $\ell$-degree Turán density. Our main result concerns the case $\ell=2$. We prove that, for every $k\ge3$, if a $k$-graph $F$ satisfies $π_2(F)=0$, then $F$ admits a $2$-vanishing order, that is, a global vertex ordering under which all edges align canonically with respect to their pairs. This extends to all uniformities a structural phenomenon previously known for $3$-graphs, and gives a higher-degree analogue of the classical fact that $π_1(F)=0$ forces $F$ to be $k$-partite. In particular, the absence of a $2$-vanishing order is a structural obstruction to vanishing $2$-degree Turán density. We also establish a suspension principle connecting consecutive degree parameters. Given a $(k-1)$-graph $F$, let $\mathcal{S}_F$ be the $k$-graph obtained by adding an apex vertex $v$ and replacing each edge $e\in E(F)$ with $v\cup e$. We show that, for $2\le \ell<k$, $π_{\ell}(\mathcal{S}_F)=0$ if and only if $π_{\ell-1}(F)=0$. This provides a bridge between different degree Turán densities and allows vanishing results to be lifted across uniformities and degree parameters. As an application, we prove that except the classical Turán density, all other degree Turán densities accumulate at zero. The proof of our main result combines random geometric building blocks, a design-theoretic gluing scheme, and random sparsification to reconcile positive $2$-degree with local vanishing structure.

math.CO↗

arXiv2026

Quasi-kernels in Hereditary Classes and Applications to Break

Recently, Nguyen, Seymour and Scott verified the small quasi-kernel conjecture for split digraphs, and initiated the study of quasi-kernels in break digraphs. Following their research, we introduce a weighted half-neighborhood property for hereditary classes of oriented graphs and show that it gives a $2n/3$ bound of small quasi-kernel for break digraphs. We also record two stronger $n/2$ results for special classes of break digraphs. Finally, using the same framework we also prove that every digraph on $n$ vertices has a quasi-kernel $Q$ with $|N_D^+[Q]|\ge \sqrt n$.

math.CO↗

arXiv2026

Large induced subgraphs with prescribed degree parity

A long-standing conjecture of Caro (Discrete Math, 1994), confirmed by Ferber and Krivelevich (Adv Math, 2022), states that every $n$-vertex graph $G$ without isolated vertices contains an induced subgraph of order linear in $n$ in which every vertex has odd degree. We generalize this result to graphs $G$ whose vertices are labeled by $\ell: V(G)\to \{0,1\}$. We require, in an induced subgraph, all $0$-labeled vertices to have even degree and all $1$-labeled vertices to have odd degree. Let $h_{\ell}(G)$ denote the maximum order of such a subgraph. Let $f_{oe}(G)=\min_{\ell} h_{\ell}(G)$ be the worst-labeling parameter. We establish a pointwise lower bound for $h_{\ell}(G)$ that immediately yields a linear lower bound in $|V(G)|$ for $f_{oe}(G)$, where $G$ has no isolated vertices. For an $n$-vertex connected graph, we obtain a sharp lower bound for $f_{oe}(G)$: $f_{oe}(G)\ge \lceil (n-1)/χ_{mm}{(G)} \rceil ,$ where $χ_{mm}{(G)}$ is the maximum chromatic number of a minor of $G.$ Using proved cases of Hadwiger's Conjecture, we show that for $t\in \{3,4,5,6\}$, if an $n$-vertex connected graph $G$ is $K_t$-minor-free, then $f_{oe}(G)\ge \lceil (n-1)/(t-1)\rceil$ and this bound is sharp for each $t\in \{3,4,5,6\}$. Finally, we conjecture that $f_{oe}(G)\ge f_o(G)/2$ for all graphs $G$ and confirm the conjecture for all trees and complete multipartite graphs.

math.CO↗

arXiv2026

A cubic refinement of Jackson's Chvátal--Erdős condition for Hamilton cycles in digraphs

For a digraph $D$, let $\aTwo(D)$ be the largest size of a vertex set no two of whose vertices lie in a common directed $2$-cycle. Let $f_2(a)$ be the least integer $K$ such that every $K$-connected digraph $D$ with $\aTwo(D)\leq a$ has a Hamilton cycle. In 1987, Jackson proved that $f_2(a)\leq 2^a(a+2)!$ and asked for better bounds, noting that a linear bound might be possible. Kühn and Osthus later observed that even a polynomial bound would be interesting. In this short note, we prove the polynomial bound $f_2(a)\leq 2a^3+2$.

math.CO↗

arXiv2026

Maximizing the Steklov eigenvalues on trees with a diameter constraint

We study the first nonzero Steklov eigenvalue $λ_2(T,δΩ)$ of the Dirichlet-to-Neumann operator on a finite tree $T$ with leaf boundary $δΩ$, under a constraint on the diameter $D$. He and Hua [Calc. Var. PDE, 2022] showed that $λ_2(T) \leq 2/D$ for any tree of diameter $D$, with the even-diameter equality case fully characterized. For odd $D$, the geometric picture underlying the sharp configurations has remained unclear beyond diameter three. We determine this picture completely for all odd diameters $D = 2r+1 \geq 5$. The sharp value of $λ_2$ is achieved on spider trees with nearly-equidistributed branch lengths, forming the family of \emph{generalized almost seesaw trees} $\mathrm{AS}(r,q+2,c,t)$, prescribed by the arithmetic of $n$ relative to $\lceil r/2 \rceil$. Together with the results of He-Hua and Lin-Zhao [Bull. Lond. Math. Soc., 2025] for even diameters and diameter three, this completes the geometric classification for every diameter. The argument is based on a scalar root equation for one-center profiles, an inverse boundary quadratic form on boundary fluxes, and a reduction scheme from arbitrary trees to two-center profiles, and then to the one-center class. The inverse variational viewpoint may be regarded as a boundary analogue of the classical distance-matrix formalism for trees initiated by Graham and Lovász [Adv. Math., 1978].

math.CO↗

arXiv2026

Majority Edge Colouring of Hypergraph

Motivated by recent work on majority edge-colourings of graphs, we initiate the study of the corresponding problem for hypergraphs. First, sharpening the probabilistic argument by a $KL$ large-deviation estimate, we obtain a sufficient minimum-degree condition of order $k^3\log(kr)$ with the sharp large-deviation constant $ I_k:=D\!\left(\frac1k\middle\|\frac1{k+1}\right)=Θ(k^{-3}), $ where $D(\cdot\|\cdot)$ denotes the binary relative entropy. Our main constructive result shows that every hypergraph of rank at most $r$ and minimum degree at least $2rk^2$ admits a $1/k$-majority $(k+1)$-edge-colouring. The proof is based on a hypergraph extension of the key discrepancy lemma used in the graph case. We also show that the logarithmic dependence on the rank can be determined asymptotically. If $μ_k(r)$ denotes the least minimum-degree threshold that guarantees a $1/k$-majority $(k+1)$-edge-colouring for all hypergraphs of rank at most $r$, then for every fixed $k\ge2$, $ μ_k(r)=\frac{\log r}{I_k}+O_k(\log\log r). $ In particular, the correct logarithmic threshold is of order $k^3\log r$. Finally, we determine the correct order of the degree--colour trade-off. For integers $k\ge2$, $p\ge1$, and $r\ge2$, let $ν_{k,p}(r)$ denote the least integer $q$ such that every hypergraph of rank at most $r$ and minimum degree at least $kp$ admits a $1/k$-majority $q$-edge-colouring. Then $ ν_{k,p}(r)=Θ_{k,p}(r^{1/p}). $ In particular, minimum degree at least $k^2-k$ guarantees a $1/k$-majority $O_k(r^{1/(k-1)})$-edge-colouring, and this exponent is best possible.

math.CO↗

arXiv2026

Effect of edge-stretching on Steklov eigenvalues and sharp Steklov eigenvalue bounds on leaf--boundary trees

Let $T$ be a finite tree with leaf set $\dO$ as the boundary and let $λ_2$ be the first nontrivial Steklov eigenvalue. Let $D$ and $\ell$ be the maximum vertex degree and the number of leaves, respectively. Motivated by the spectral influence of neck-stretching on Riemannian manifolds, we investigate a discrete counterpart--edge-stretching--and its effect on the Steklov eigenvalues of graphs. We prove that Steklov eigenvalues decrease monotonically under the edge--stretching operation. As a consequence, we prove that $λ_2\le D/\ell$, with equality if and only if $T$ is a star. This fundamentally improves the constant in He--Hua's bound $λ_2\le 4(D-1)/\ell$ to the optimal value~$1$. We also provide a closed-form diagonalization of the Steklov problem on level--regular trees, yielding explicit eigenvalues and multiplicities. In addition, we provide a general upper bound $λ_k\le \min\{1,\,16Dk/\ell\}$ for higher eigenvalues. Systematic numerical experiments verify the sharp bound and provide evidence for the extremal conjecture of Lin--Zhao on balanced minimum--height trees.

math.CO↗

arXiv2026

A weighted cycle-localization inequality

In 1959, Erdős and Gallai showed that every $2$-connected graph $G$ contains a cycle of length at least $\frac{2|E(G)|}{|V(G)|-1}$. This result was subsequently extended to weighted graphs by Bondy and Fan in 1991. A natural local variant of this problem arises by considering, for each edge $e\in E(G)$, the quantity $c(e)$, defined as the length of the longest cycle in $G$ containing $e$ (with $c(e)=2$ if $e$ is a bridge). Zhao and Zhang recently proved that for every graph $G$ on $n$ vertices satisfies $\sum_{e\in E(G)}\frac{1}{c(e)}\le \frac{n-1}{2}.$ In this note, we establish a weighted generalization of this inequality. For a weighted graph $(G,w)$ with positive edge weights, let $C_w(e)$ denote the maximum weight of a cycle containing $e$ (setting $C_w(e)=2w(e)$ if $e$ is a bridge). We prove that $$ \sum_{e\in E(G)}\frac{w(e)}{C_w(e)}\le \frac{n-1}{2}. $$ Our result can be viewed as a weighted local analogue of the Bondy-Fan theorem, thereby establishing a correspondence between the global and local perspectives. Furthermore, we present a broad class of graphs attaining equality and derive necessary conditions for equality.

math.CO↗

arXiv2026

Refinements of Alon-Babai-Suzuki-type intersection theorems via non-shadows and binomial support

We prove a multilevel non-shadow refinement of the Alon--Babai--Suzuki (ABS) nonuniform restricted-intersection theorem. Let $K=\{k_1,\dots,k_r\}$ and let $L$ be a set with $|L|=s$. If $\mathcal{F}\subseteq \bigcup_{k\in K}\binom{[n]}{k}$ is $L$-intersecting and $k_i>s-r$ for every $i$, then $|\mathcal{F}| + \sum_{j=s-r+1}^{s} |\mathcal{N}_j(\mathcal{F})| \le N(n,s,r),$ equivalently $|\mathcal{F}| \le \sum_{j=s-r+1}^{s} |\partial_j\mathcal{F}|.$ Thus the ABS bound is sharpened by the total non-shadow deficit on the top $r$ levels. In the modular setting, we take a coefficient-sensitive viewpoint: the polynomial method depends not just on the degree of the annihilator polynomial $P_L(t)=\prod_{\ell\in L}(t-\ell)\in\mathbb{F}_p[t]$, but on which binomial terms actually appear in it. This yields a gap-free modular bound depending only on the active support levels of $P_L$. For almost-initial residue patterns $L=\{0,1,\dots,s-m-1\}\cup R \pmod p$ we obtain the collapse $|\mathcal{F}|\le \sum_{i=0}^{m}\binom{n}{s-i}.$ In particular, for consecutive residues $L=\{0,1,\dots,s-1\}\pmod p$ we get the sharp bound $|\mathcal{F}|\le \binom{n}{s}$, giving a partial negative answer to a question of Alon--Babai--Suzuki: the modular ABS bound $N(n,s,r)$ is not attainable in the consecutive-residue regime whenever $r\ge 2$.

math.CO↗