arXiv subjects

Yantao Tang

Publications and source records attributed to Yantao Tang.

6 recordsLinked to original sources

arXiv2026

Counting Near-Spanning Matchings in Latin Squares and Steiner Triple Systems

Montgomery recently proved that for sufficiently large $n$, every Latin square of order $n$ has a partial transversal with $n-1$ cells, and every Steiner triple system of order $n$ has a matching with $\lfloor n/3\rfloor-1$ edges, thus confirming the Ryser--Brualdi--Stein conjecture for even $n$ and the conjecture of Brouwer. We prove sharp enumerative refinements of these results: there is an absolute constant $c>0$ such that, for sufficiently large $n$, 1) every Latin square of order $n$ has $ \left((1\pm n^{-c})\frac{n}{\mathrm {e}^2}\right)^n$ partial transversals with $n-1$ cells; 2) every Steiner triple system of order $n$ has $ \left((1\pm n^{-c})\frac{n}{2\mathrm {e}^2}\right)^{\lfloor n/3\rfloor}$ matchings with $\lfloor n/3\rfloor-1$ edges. The first estimate confirms predictions of Montgomery and Kelly.

math.CO↗

arXiv2026

Tripartite Zarankiewicz numbers and norm graphs

For fixed integers $s\ge t\ge2$, let $\operatorname{ex}(n,n,n,K_{s,t})$ denote the maximum number of edges in a tripartite $K_{s,t}$-free graph with $n$ vertices in each part. When $s\ge(t-1)!+1$, let $r$ be the largest integer satisfying $s\ge(t-1)!r^{t-1}+1$. Using the quotient norm graphs of Alon, R\'onyai and Szab\'o, we prove that \[ \operatorname{ex}(n,n,n,K_{s,t}) \ge \left(\frac{3}{2^{1/t}}r^{1-1/t}+o(1)\right)n^{2-1/t}. \] Improving an upper bound of Tait and Timmons, we prove that, for all $s\ge t\ge 2$, \[ \operatorname{ex}(n,n,n,K_{s,t})\le \left(\frac{3}{2^{1/t}}(s-t+1)^{1/t}+o(1)\right)n^{2-1/t}. \] Together, these bounds recover the results for $t=2$, and give the new asymptotic formula \[ \operatorname{ex}(n,n,n,K_{3,3}) =\left(\frac{3}{\sqrt[3]{2}}+o(1)\right)n^{5/3}. \] Analogous results extend to $k$-partite graphs containing no $K_{s, t}$ whose $s$-vertex or $t$-vertex side lies in a single part. As an application of our tripartite construction, we determine the tripartite multicolor Ramsey number of $K_{3,3}$ asymptotically.

math.CO↗

arXiv2025

Number of independent transversals in multipartite graphs

An independent transversal in a multipartite graph is an independent set that intersects each part in exactly one vertex. We show that for every even integer $r\ge 2$, there exist $c_r>0$ and $n_0$ such that every $r$-partite graph with parts of size $n\ge n_0$ and maximum degree at most $rn/(2r-2)-t$, where $t=o(n)$, contains at least $c_r t n^{r-1}$ independent transversals. This is best possible up to the value of $c_r$. Our result confirms a conjecture of Haxell and Szab\'o from 2006 and partially answers a question raised by Erd\H{o}s in 1972 and studied by Bollob\'as, Erd\H{o}s and Szemer\'edi in 1975. We also show that, given any integer $s\ge 2$ and even integer $r\ge 2$, there exist $c_{r,s}>0$ and $n_0$ such that every $r$-partite graph with parts of size $n\ge n_0$ and maximum degree at most $rn/(2r-2)- c_{r, s} n^{1-1/s}$ contains an independent set with exactly $s$ vertices in each part. This is best possible up to the value of $c_{r, s}$ if a widely believed conjecture for the Zarankiewicz number holds. Our result partially answers a question raised by Di Braccio and Illingworth recently.

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arXiv2024

Extremal density for subdivisions with length or sparsity constraints

Given a graph $H$, a balanced subdivision of $H$ is obtained by replacing all edges of $H$ with internally disjoint paths of the same length. In this paper, we prove that for any graph $H$, a linear-in-$e(H)$ bound on average degree guarantees a balanced $H$-subdivision. This strengthens an old result of Bollob\'as and Thomason, and resolves a question of Gil-Fern\'{a}ndez, Hyde, Liu, Pikhurko and Wu. We observe that this linear bound on average degree is best possible whenever $H$ is logarithmically dense. We further show that this logarithmic density is the critical threshold: for many graphs $H$ below this density, its subdivisions are forcible by a sublinear-in-$e(H)$ bound on average degree. We provide such examples by proving that the subdivisions of any almost bipartite graph $H$ with sublogarithmic density are forcible by a sublinear-in-$e(H)$ bound on average degree, provided that $H$ satisfies some additional separability condition.

math.CO↗

arXiv2023

On powers of Hamilton cycles in Ramsey-Tur\'{a}n Theory

We prove that for $r\in \mathbb{N}$ with $r\geq 2$ and $\mu>0$, there exist $\alpha>0$ and $n_{0}$ such that for every $n\geq n_{0}$, every $n$-vertex graph $G$ with $\delta(G)\geq \left(1-\frac{1}{r}+\mu\right)n$ and $\alpha(G)\leq \alpha n$ contains an $r$-th power of a Hamilton cycle. We also show that the minimum degree condition is asymptotically sharp for $r=2, 3$ and the $r=2$ case was recently conjectured by Staden and Treglown.

math.CO↗

arXiv2022

Balanced subdivisions of cliques in graphs

Given a graph $H$, a balanced subdivision of $H$ is a graph obtained from $H$ by subdividing every edge the same number of times. In 1984, Thomassen conjectured that for each integer $k\ge 1$, high average degree is sufficient to guarantee a balanced subdivision of $K_k$. Recently, Liu and Montgomery resolved this conjecture. We give an optimal estimate up to an absolute constant factor by showing that there exists $c>0$ such that for sufficiently large $d$, every graph with average degree at least $d$ contains a balanced subdivision of a clique with at least $cd^{1/2}$ vertices. It also confirms a conjecture from Verstra{\"e}te: every graph of average degree $cd^2$, for some absolute constant $c>0$, contains a pair of disjoint isomorphic subdivisions of the complete graph $K_d$. We also prove that there exists some absolute $c>0$ such that for sufficiently large $d$, every $C_4$-free graph with average degree at least $d$ contains a balanced subdivision of the complete graph $K_{cd}$, which extends a result of Balogh, Liu and Sharifzadeh.

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