arXiv subjects

Zhifei Yan

Publications and source records attributed to Zhifei Yan.

13 recordsLinked to original sources

arXiv2026

Exponential tails for factors and the chromatic number of random graphs

The celebrated result of Johansson, Kahn and Vu determined the threshold order for clique factors in random graphs, and subsequent work identified the sharp threshold and the corresponding hitting-time phenomenon. In this paper we study the probability that there is no $K_r$-factor above the threshold and, more generally, the probability that the largest $K_r$-matching covers less than $n-s$ vertices of $G(n,p)$. For every fixed $r\ge3$, throughout the range $$n^{-2/r}(\log n)^{1/\binom r2}\ll p\ll n^{-2/(r+1)},\qquad n-s\in r\mathbb Z,\qquad s=o(n),$$ we prove $$\mathbb P\bigl(ϕ_r^s(G(n,p))=0\bigr)=\exp\left(-Θ_r\!\left((s+1)\frac{μ_r(n,p)}n\right)\right),$$ where $ϕ_r^s(G)$ is the number of $K_r$-matchings covering exactly $n-s$ vertices and $μ_r(n,p):=\binom nrp^{\binom r2}$. The lower bound is given by $s+1$ vertices which lie in no copy of $K_r$. For the upper bound we develop an iterable one-root version of the Johansson--Kahn--Vu method. As a structural consequence, we show that the remainder of $G(n,p)$ outside every maximal $K_r$-matching has an almost-perfect $K_{r-1}$-matching throughout the sparse clique window. Independently, we prove a central limit theorem for the maximum $K_r$-matching number. Combining these inputs and a structural theorem for $r=2$ from our earlier work, we prove a central limit theorem for the chromatic number of very dense random graphs: for every $r\ge2$ and $n^{-2/r}(\log n)^{1/\binom r2}\ll p\ll n^{-2/(r+1)},$ $$\frac{χ(G(n,1-p))-\mathbb Eχ(G(n,1-p))}{\sqrt{μ_{r+1}(n,p)}/r}\xrightarrow{\mathrm d}\mathcal N(0,1),\qquad\operatorname{Var}\bigl(χ(G(n,1-p))\bigr) \sim\frac{μ_{r+1}(n,p)}{r^2}.$$ This settles the Surya--Warnke conjecture throughout the interior of every clique window with $r\ge2$, strengthening its concentration prediction to a Gaussian limit with asymptotically exact variance.

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arXiv2026

Structural Reductions for Monochromatic Matchings and Ramsey Tilings

The Alon--Frankl--Lovász theorem determines the chromatic number of Kneser hypergraphs; equivalently, it gives the sharp minimum size of a monochromatic matching in every edge-colouring of a complete uniform hypergraph. Its known general proofs are topological. We introduce a topology-free structural framework. It reduces every colouring of a pseudorandom $t$-graph, with only $o(n)$ loss in the largest monochromatic matching, to a colouring of $K_n^{(t)}$ whose vertex set has at most $r$ parts and whose edge colours depend only on intersection profiles. Together with a stability analysis at the critical scale, we prove an exact robust form: there exists $c=c(r,t)>0$ such that, if a $t$-graph satisfies $δ(\mathcal G)\ge(1-c)\binom{n-1}{t-1}$, then every $r$-colouring of $\mathcal G$ contains a monochromatic matching of the exact optimal size for all sufficiently large $n$. This gives a topology-free proof of the Alon--Frankl--Lovász theorem for large $n$, a sparse random transference theorem, and the exact value predicted by Meunier's stable Kneser conjecture throughout the range covered by our robust AFL theorem. We further develop the framework for Ramsey graph tilings. For a graph $H$, let $Rt_r(H;K_n)$ be the minimum, over all $r$-edge-colourings of $K_n$, of the largest monochromatic $H$-tiling. We prove $$ Rt_r(H;K_n)=(β_{r,H}+o(1))n, $$ where $β_{r,H}$ is effectively computable from finitely many rational linear programs depending only on $H$ and $r$. An additional multipartite Ramsey argument is needed to reconstruct a consistent coloured template. This gives an effective asymptotic solution to the multicolour Ramsey-tiling problem, extending the classical two-colour theorem of Burr, Erdős and Spencer. We also determine explicit constants for several natural families.

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arXiv2026

A rainbow version of Lehel's conjecture

Lehel's conjecture states that every 2-edge-colouring of K_n admits a partition of its vertex set into two monochromatic cycles. It was proven for sufficiently large n by Łuczak, Rödl, and Szemerédi in 1998, later improved by Allen in 2008, and fully resolved by Bessy and Thomassé in 2010. In this paper, we consider a rainbow analogue of Lehel's conjecture in the setting of properly edge-coloured complete graphs. We prove that, for sufficiently large n, every properly edge-coloured Kn admits a partition of its vertex set into two vertex-disjoint rainbow cycles

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arXiv2026

Tight Staircase Bounds for Cyclic Subsets below Dirac's Threshold

Let $\operatorname{Cyc}(G)$ denote the number of cyclic subsets in a graph $G$, which are subsets that induce a Hamiltonian subgraph. Draganić, Keevash and Müyesser recently proved that every regular Dirac graph has $Ω(2^n)$ cyclic subsets, resolving a problem of Erdős and Faudree. We determine the sharp asymptotic lower bound throughout the linear range below Dirac's threshold. Let $G$ be an $n$-vertex $d$-regular graph with $d=Ω(n)$ and $d<n/2$, then $$ \operatorname{Cyc}(G)\ge (q-o(1))2^{n/q}, \quad \text{where } \quad q=\left\lfloor \frac{n}{d+1}\right\rfloor \ge 2. $$ This bound is asymptotically best possible, including the leading coefficient $q$, as witnessed at the staircase levels by the disjoint union of $q$ equal cliques. Consequently, the optimal exponential rate changes by discrete jumps as $d$ crosses the thresholds $n/k$, rather than varying smoothly with $d$. We also prove the optimal exponential rate at the Dirac boundary: every $n$-vertex $n/2$-regular graph satisfies $\operatorname{Cyc}(G)\ge 2^{(1-o(1))n},$ which is sharp up to a subexponential factor by $K_{n/2,n/2}$.

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arXiv2026

The sharp threshold for rainbow stackings of random edge-colourings

A rainbow stacking of $m$ independent, uniformly random $r$-edge-colourings of $K_n$ is a tuple of vertex permutations that superimposes the colourings such that no two edges of the same colour overlap. The study of the critical palette size $r$ required for the existence of such stackings was recently initiated by Alon, Defant, and Kravitz [Bull. Lond. Math. Soc., 57, 2025], who bounded the phase transition within a constant-order window around $\frac{m\binom{n}{2}}{2\log(n!)}$. We determine the constant term in this transition. For every fixed $m\ge2$ and every function $ω(n)\to\infty$, with high probability there is no rainbow stacking if $$r\le \frac{m\binom{n}{2}}{2\log(n!)}+\frac{2m-1}{6}-\frac{ω(n)}{(\log n)^2},$$ while with high probability there is one if $$r\ge \frac{m\binom{n}{2}}{2\log(n!)}+\frac{2m-1}{6}+\frac{ω(n)}{(\log n)^2}.$$ Our proof combines a chromatic-polynomial expansion for an auxiliary conflict graph with a refined estimate of the associated weighted permutation sum. Our result yields the exact threshold $\Big\lceil \frac{m\binom{n}{2}}{2\log(n!)}+\frac{2m-1}{6}\Big\rceil$ for a density-one set of integers $n$, resolving a problem of Alon, Defant and Kravitz.

math.CO↗

arXiv2026

Constructor--Blocker games forbidding even cycles

The Constructor--Blocker game is played on the edge set of $K_n$. Two players alternately claim previously unclaimed edges. Constructor aims to maximize the number of copies of a target graph $H$ in her graph while keeping it $F$-free throughout the game, whereas Blocker aims to minimize this number. When both players play optimally and Constructor moves first, the final number of copies of $H$ in Constructor's graph is called the score of the game and is denoted by $g(n,H,F)$. Recently, Balogh, Chen, and English systematically studied this game for non-bipartite forbidden graphs $F$. However, bipartite forbidden graphs present additional difficulties. In this paper, we focus on the case in which the forbidden graph is an even cycle. First, using a finite-field-geometric construction, we confirm a conjecture of Balogh, Chen and English by proving \[ g(n,K_3,C_4)=Θ(n^{3/2}). \] We also establish two-sided bounds for $g(n,K_3,C_{2k})$ that relate the game score to classical extremal numbers. Our proofs reveal a structural distinction between the cases $k=2$ and $k\ge3$: a vertex-duplication method works for longer even cycles, but it necessarily creates $4$-cycles and therefore cannot be used in the $C_4$-free game. Finally, we determine the exact order of $g(n,C_t,C_{2k})$ for all $k\ge2$ and all $t\ge4$.

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arXiv2026

Triangle packings in randomly perturbed graphs

The longstanding Nash-Williams conjecture asserts that every $K_3$-divisible graph $G$ with $δ(G)\ge 3n/4$ admits a triangle decomposition. In the random setting, Frankl and Rödl showed that, with high probability, $G(n,p)$ contains a triangle packing covering all but $o(n^2p)$ edges whenever $p\ge n^{-1/2+\varepsilon}$. In this paper, we study near-perfect triangle packings in randomly perturbed graphs. We prove that for every $d>0$ and every $p>2d/(1+2d)$, if $G_d$ is a $dn$-regular graph on $n$ vertices, then with high probability the union $G_d\cup G(n,p)$ contains a triangle packing covering all but $o(n^2)$ edges. Moreover, this bound on $p$ is best possible for $0<d\le 1/2$, thereby determining the threshold in this range. A key ingredient in the proof is a new triangle-weighting lemma for weighted complete graphs.

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arXiv2026

Sharp threshold for Hamilton cycles in randomly perturbed sparse graphs

We determine the sharp threshold for Hamilton cycles in randomly perturbed sparse graphs. For any $α=α(n)=o(1)$, let $G_α$ be an $n$-vertex graph with minimum degree $δ(G_α)\geαn$. We prove that if $$p\ge(1+\varepsilon)\frac{\log(1/α)}{n},$$ then the union $G_α\cup G(n,p)$ is Hamiltonian asymptotically almost surely. This significantly strengthens a recent result of Hahn-Klimroth, Maesaka, Mogge, Mohr, and Parczyk by improving the leading constant from 6 to the optimal value of 1. Crucially, we show that this bound on $p$ is best possible when $αn\rightarrow\infty$, thereby establishing the exact probability threshold for Hamiltonicity in this sparse regime. Our proof relies on a robust random expansion lemma, Pósa's booster lemma, and a sprinkling argument.

math.CO↗

arXiv2026

Ramsey properties for tilings in random graphs

Let $mH$ be the graph formed by $m$ vertex-disjoint copies of a graph $H$. Let $G \to (H)_r$ denote that, in any $r$-colouring of the edges of $G$, there exists a monochromatic copy of $H$. In 1975, Burr, Erdős, and Spencer showed that if $H$ is a graph on $k$ vertices whose independence number is $α$, then $K_n \to (mH)_2$, where $m\sim n/(2k-α)$, and that the $1/(2k-α)$ factor is best possible. In the 1990s, Rödl and Ruciński proved that, for all but a few graphs~$H$, the threshold for the property $\mathbb{G}(n,p) \to (H)_r$ is $n^{-1/m_2(H)}$. In this paper, generalizing the result of Burr, Erdős, and Spencer, we prove that $n^{-1/\max\{m_2(H),1\}}$ is the threshold for the property $\mathbb{G}(n,p) \to (mH)_2$, where $m\sim n/(2k-α)$. This threshold matches the one found by Rödl and Ruciński for most graphs $H$, extending their result in the case $r=2$.

math.CO↗

arXiv2026

Colour diversity in spanning structures under Dirac-type conditions

Finding spanning structures with many distinct colours in properly edge-coloured graphs is a central theme in extremal combinatorics. A classical result of Andersen shows that every proper edge-colouring of the complete graph $K_n$ contains a Hamilton cycle with $n - O(n^{1/2})$ distinct colours. In the bipartite setting, the analogous question for perfect matchings is closely related to permutations in Latin squares. In this paper, we investigate how a Dirac-type minimum degree condition forces colour diversity in spanning structures. For every constant $1/2 < c \le 1$, we prove the following. $\bullet$ Every properly edge-coloured graph $G$ on $n$ vertices with $δ(G)\ge cn$ contains a Hamilton cycle with at least $cn - O(n^{1/2})$ distinct colours. $\bullet$ Every subset of an $n\times n$ Latin square with at least $cn$ cells in each row and each column contains a permutation with at least $cn - O(n^{2/3})$ distinct symbols. Both bounds are best possible up to the error term.

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arXiv2025

Colour-biased Hamilton cycles in randomly perturbed graphs

Given a graph $G$ and an $r$-edge-colouring $χ$ on $E(G)$, a Hamilton cycle $H\subset G$ is said to have $t$ colour-bias if $H$ contains $n/r+t$ edges of the same colour in $χ$. Freschi, Hyde, Lada and Treglown showed every $r$-coloured graph $G$ on $n$ vertices with $δ(G)\geq(r+1)n/2r+t$ contains a Hamilton cycle $H$ with $Ω(t)$ colour-bias, generalizing a result of Balogh, Csaba, Jing and Pluhár. In 2022, Gishboliner, Krivelevich and Michaeli proved that the random graph $G(n,m)$ with $m\geq(1/2+\varepsilon)n\log n$ typically admits an $Ω(n)$ colour biased Hamilton cycle in any $r$-colouring. In this paper, we investigate colour-biased Hamilton cycles in randomly perturbed graphs. We show that for every $α>0$, adding $m=O(n)$ random edges to a graph $G_α$ with $δ(G_α)\geq αn$ typically ensures a Hamilton cycle with $Ω(n)$ colour bias in any $r$-colouring of $G_α\cup G(n,m)$. Conversely, for certain $G_α$, reducing the number of random edges to $m=o(n)$ may eliminate all colour biased Hamilton cycles of $G(n,m)\cup G$ in a certain colouring. In contrast, at the critical endpoint $α=(r+1)/2r$, adding $m$ random edges typically results in a Hamilton cycle with $Ω(m)$ colour-bias for any $1\ll m\leq n$.

math.CO↗

arXiv2024

Near rainbow Hamilton cycles in dense graphs

Finding near-rainbow Hamilton cycles in properly edge-coloured graphs was first studied by Andersen, who proved in 1989 that every proper edge colouring of the complete graph on $n$ vertices contains a Hamilton cycle with at least $n-\sqrt{2n}$ distinct colours. This result was improved to $n-O(\log^2 n)$ by Balogh and Molla in 2019. In this paper, we consider Anderson's problem for general graphs with a given minimum degree. We prove every globally $n/8$-bounded (i.e. every colour is assigned to at most $n/8$ edges) properly edge-coloured graph $G$ with $δ(G) \geq (1/2+\varepsilon)n$ contains a Hamilton cycle with $n-o(n)$ distinct colours. Moreover, we show that the constant $1/8$ is best possible.

math.CO↗

arXiv2024

The chromatic number of very dense random graphs

The chromatic number of a very dense random graph $G(n,p)$, with $p \ge 1 - n^{-c}$ for some constant $c > 0$, was first studied by Surya and Warnke, who conjectured that the typical deviation of $χ(G(n,p))$ from its mean is of order $\sqrt{μ_r}$, where $μ_r$ is the expected number of independent sets of size $r$, and $r$ is maximal such that $μ_r > 1$, except when $μ_r = O(\log n)$. They moreover proved their conjecture in the case $n^{-2} \ll 1 - p = O(n^{-1})$. In this paper, we study $χ(G(n,p))$ in the range $n^{-1}\log n \ll 1 - p \ll n^{-2/3}$, that is, when the largest independent set of $G(n,p)$ is typically of size 3. We prove in this case that $χ(G(n,p))$ is concentrated on some interval of length $O(\sqrt{μ_3})$, and for sufficiently `smooth' functions $p = p(n)$, that there are infinitely many values of $n$ such that $χ(G(n,p))$ is not concentrated on any interval of size $o(\sqrt{μ_3})$. We also show that $χ(G(n,p))$ satisfies a central limit theorem in the range $n^{-1} \log n \ll 1 - p \ll n^{-7/9}$.

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