arXiv subjects

Dingjia Mao

Publications and source records attributed to Dingjia Mao.

6 recordsLinked to original sources

arXiv2026

Pancyclicity of graphs perturbed by a random $F$-factor

We determine the sharp minimum-degree threshold for Hamiltonicity in graphs perturbed by a uniformly random $K_r$-factor, resolving a conjecture of Espuny Díaz and Girão [Random Structures Algorithms, 2023]. In fact, we prove the stronger pancyclic statement. Let $α^*(K_r)$ and $α_{\text{pan}}^*(K_r)$ denote the Hamiltonicity and pancyclicity thresholds, respectively. We show that $α^*(K_r)=α_{\text{pan}}^*(K_r)=ρ_r$, where $ρ_r$ is the unique positive solution of $x^r+rx-1=0$. The proof is obtained from a general framework for perturbations by a uniformly random $F$-factor, where $F$ is an arbitrary fixed connected graph.

math.CO↗

arXiv2025

Hamilton cycles in regular graphs perturbed by a random 2-factor

In this paper, we prove that for each $d \geq 2$, the union of a $d$-regular graph with a uniformly random $2$-factor on the same vertex set is Hamiltonian with high probability. This resolves a conjecture by Draganić and Keevash for all values of $d$.

math.CO↗

arXiv2024

Hamiltonicity of Sparse Pseudorandom Graphs

We show that every $(n,d,λ)$-graph contains a Hamilton cycle for sufficiently large $n$, assuming that $d\geq \log^{6}n$ and $λ\leq cd$, where $c=\frac{1}{70000}$. This significantly improves a recent result of Glock, Correia and Sudakov, who obtained a similar result for $d$ that grows polynomially with $n$. The proof is based on a new result regarding the second largest eigenvalue of the adjacency matrix of a subgraph induced by a random subset of vertices, combined with a recent result on connecting designated pairs of vertices by vertex-disjoint paths in $(n,d,λ)$-graphs. We believe that the former result is of independent interest and will have further applications.

math.CO↗

arXiv2024

Dirac-type Problem of Rainbow matchings and Hamilton cycles in Random Graphs

Given a family of graphs $G_1,\dots,G_{n}$ on the same vertex set $[n]$, a rainbow Hamilton cycle is a Hamilton cycle on $[n]$ such that each $G_c$ contributes exactly one edge. We prove that if $G_1,\dots,G_{n}$ are independent samples of $G(n,p)$ on the same vertex set $[n]$, then for each $\varepsilon>0$, whp, every collection of spanning subgraphs $H_c\subseteq G_c$, with $δ(H_c)\geq(\frac{1}{2}+\varepsilon)np$, admits a rainbow Hamilton cycle. A similar result is proved for rainbow perfect matchings in a family of $n/2$ graphs on the same vertex set $[n]$.

math.CO↗

arXiv2024

Regular bipartite decompositions of pseudorandom graphs

In 1972, Kotzig proved that for every even $n$, the complete graph $K_n$ can be decomposed into $\lceil\log_2n\rceil$ edge-disjoint regular bipartite spanning subgraphs, which is best possible. In this paper, we study regular bipartite decompositions of $(n,d,λ)$-graphs, where $n$ is an even integer and $d_0\leq d\leq n-1$ for some absolute constant $d_0$. With a randomized algorithm, we prove that such an $(n,d,λ)$-graph with $λ\leq d/12$ can be decomposed into at most $\log_2 d + 36$ regular bipartite spanning subgraphs. This is best possible up to the additive constant term. As a consequence, we also improve the best known bounds on $λ= λ(d)$ by Ferber and Jain (2020) to guarantee that an $(n,d,λ)$-graph on an even number of vertices admits a $1$-factorization, showing that $λ\leq cd$ is sufficient for some absolute constant $c > 0$.

math.CO↗

arXiv2021

A stability result on matchings in 3-uniform hypergraphs

Let $n,s,k$ be three positive integers such that $1\leq s\leq(n-k+1)/k$ and let $[n]=\{1,\ldots,n\}$. Let $H$ be a $k$-graph with vertex set $\{1,\ldots,n\}$, and let $e(H)$ denote the number of edges of $H$. Let $ν(H)$ and $τ(H)$ denote the size of a largest matching and the size of a minimum vertex cover in $H$, respectively. Define $A^k_i(n,s):=\{e\in\binom{[n]}{k}:|e\cap[(s+1)i-1]|\geq i\}$ for $2\leq i\leq k$ and $HM^k_{n,s}:=\big\{e\in\binom{[n]}{k}:e\cap[s-1]\neq\emptyset\big\} \cup\big\{S\big\}\cup \big\{e\in\binom{[n]}{k}: s\in e, e\cap S\neq \emptyset\}$, where $S=\{s+1,\ldots,s+k\}$. Frankl and Kupavskii conjectured that if $ν(H)\leq s$ and $τ(H)>s$, then $e(H)\leq \max\{|A^k_2(n,s)|,\ldots ,|A^k_k(n,s)|,|HM^k_{n,s}|\}$. In this paper, we prove this conjecture for $k=3$ and sufficiently large $n$.

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