arXiv subjects

Hongliang Lu

Publications and source records attributed to Hongliang Lu.

At least 73 records · Page 4Linked to original sources

arXiv2022

Improved Bound on Vertex Degree Version of Erdős Matching Conjecture

For a $k$-uniform hypergraph $H$, let $δ_1(H)$ denote the minimum vertex degree of $H$, and $ν(H)$ denote the size of the largest matching in $H$. In this paper, we show that for any $k\geq 3$ and $β>0$, there exists an integer $n_0(β,k)$ such that for positive integers $n\geq n_0$ and $m\leq (\frac{k}{2(k-1)}-β)\frac{n}{k}$, if $H$ is an $n$-vertex $k$-graph with $δ_1(H)>{{n-1}\choose {k-1}}-{{n-m}\choose {k-1}},$ then $ν(H)\geq m$. This improves upon earlier results of Bollobás, Daykin and Erdős (1976) for the range $n> 2k^3(m+1)$ and Huang and Zhao (2017) for the range $n\geq 3k^2 m$.

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arXiv2022

Spectral radius and rainbow matchings of graphs

Let $n,m$ be integers such that $1\leq m\leq (n-2)/2$ and let $[n]=\{1,\ldots,n\}$. Let $\mathcal{G}=\{G_1,\ldots,G_{m+1}\}$ be a family of graphs on the same vertex set $[n]$. In this paper, we prove that if for any $i\in [m+1]$, the spectral radius of $G_i$ is not less than $\max\{2m,\frac{1}{2}(m-1+\sqrt{(m-1)^2+4m(n-m)})\}$, then $\mathcal{G}$ admits a rainbow matching, i.e. a choice of disjoint edges $e_i\in G_i$, unless $G_1=G_2=\ldots=G_{m+1}$ and $G_1\in \{K_{2m+1}\cup (n-2m-1)K_1, K_m\vee (n-m)K_1\}$.

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arXiv2022

Toughness, hamiltonicity and spectral radius in graphs

The study of the existence of hamiltonian cycles in a graph is a classic problem in graph theory. By incorporating toughness and spectral conditions, we can consider Chvátal's conjecture from another perspective: what is the spectral condition to guarantee the existence of a hamiltonian cycle among $t$-tough graphs? We first give the answer to $1$-tough graphs, i.e. if $ρ(G)\geqρ(M_{n})$, then $G$ contains a hamiltonian cycle, unless $G\cong M_{n}$, where $M_{n}=K_{1}\nabla K_{n-4}^{+3}$ and $K_{n-4}^{+3}$ is the graph obtained from $3K_{1}\cup K_{n-4}$ by adding three independent edges between $3K_{1}$ and $K_{n-4}$. The Brouwer's toughness theorem states that every $d$-regular connected graph always has $t(G)>\frac{d}λ-1$ where $λ$ is the second largest absolute eigenvalue of the adjacency matrix. In this paper, we extend the result in terms of its spectral radius, i.e. we provide a spectral condition for a graph to be 1-tough with minimum degree $δ$ and to be $t$-tough, respectively.

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arXiv2021

Spectral radius and $[a,b]$-factors in graphs

An $[a,b]$-factor of a graph $G$ is a spanning subgraph $H$ such that $a\leq d_{H}(v)\leq b$ for each $v\in V(G)$. In this paper, we provide spectral conditions for the existence of an odd $[1,b]$-factor in a connected graph with minimum degree $δ$ and the existence of an $[a,b]$-factor in a graph, respectively. Our results generalize and improve some previous results on perfect matchings of graphs. For $a=1$, we extend the result of O\cite{S.O} to obtain an odd $[1,b]$-factor and further improve the result of Liu, Liu and Feng\cite{W.L} for $a=b=1$. For $n\geq 3a+b-1$, we confirm the conjecture of Cho, Hyun, O and Park\cite{E.C}. We conclude some open problems in the end.

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arXiv2021

Co-degree threshold for rainbow perfect matchings in uniform hypergraphs

Let $k$ and $n$ be two integers, with $k\geq 3$, $n\equiv 0\pmod k$, and $n$ sufficiently large. We determine the $(k-1)$-degree threshold for the existence of a rainbow perfect matchings in $n$-vertex $k$-uniform hypergraph. This implies the result of Rödl, Ruciński, and Szemerédi on the $(k-1)$-degree threshold for the existence of perfect matchings in $n$-vertex $k$-uniform hypergraphs. In our proof, we identify the extremal configurations of closeness, and consider whether or not the hypergraph is close to the extremal configuration. In addition, we also develop a novel absorbing device and generalize the absorbing lemma of Rödl, Ruciński, and Szemerédi.

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arXiv2021

On the rainbow matching conjecture for 3-uniform hypergraphs

Aharoni and Howard, and, independently, Huang, Loh, and Sudakov proposed the following rainbow version of Erdős matching conjecture: For positive integers $n,k,m$ with $n\ge km$, if each of the families $F_1,\ldots, F_m\subseteq {[n]\choose k}$ has size more than $\max\{\binom{n}{k} - \binom{n-m+1}{k}, \binom{km-1}{k}\}$, then there exist pairwise disjoint subsets $e_1,\dots, e_m$ such that $e_i\in F_i$ for all $i\in [m]$. We prove that there exists an absolute constant $n_0$ such that this rainbow version holds for $k=3$ and $n\geq n_0$. We convert this rainbow matching problem to a matching problem on a special hypergraph $H$. We then combine several existing techniques on matchings in uniform hypergraphs: find an absorbing matching $M$ in $H$; use a randomization process of Alon et al. to find an almost regular subgraph of $H-V(M)$; and find an almost perfect matching in $H-V(M)$. To complete the process, we also need to prove a new result on matchings in 3-uniform hypergraphs, which can be viewed as a stability version of a result of Łuczak and Mieczkowska and might be of independent interest.

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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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arXiv2021

Rainbow perfect matchings for 4-uniform hypergraphs

Let $n$ be a sufficiently large integer with $n\equiv 0\pmod 4$ and let $F_i \subseteq{[n]\choose 4}$ where $i\in [n/4]$. We show that if each vertex of $F_i$ is contained in more than ${n-1\choose 3}-{3n/4\choose 3}$ edges, then $\{F_1, \ldots ,F_{n/4}\}$ admits a rainbow matching, i.e., a set of $n/4$ edges consisting of one edge from each $F_i$. This generalizes a deep result of Khan on perfect matchings in 4-uniform hypergraphs.

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arXiv2021

A note on exact minimum degree threshold for fractional perfect matchings

Rödl, Ruciński, and Szemerédi determined the minimum $(k-1)$-degree threshold for the existence of fractional perfect matchings in $k$-uniform hypergrahs, and Kühn, Osthus, and Townsend extended this result by asymptotically determining the $d$-degree threshold for the range $k-1>d\ge k/2$. In this note, we prove the following exact degree threshold: Let $k,d$ be positive integers with $k\ge 4$ and $k-1>d\geq k/2$, and let $n$ be any integer with $n\ge k^2$. Then any $n$-vertex $k$-uniform hypergraph with minimum $d$-degree $δ_d(H)>{n-d\choose k-d} -{n-d-(\lceil n/k\rceil-1)\choose k-d}$ contains a fractional perfect matching. This lower bound on the minimum $d$-degree is best possible. We also determine optimal minimum $d$-degree conditions which guarantees the existence of fractional matchings of size $s$, where $0<s\le n/k$ (when $k/2\le d\le k-1$), or with $s$ large enough and $s\le n/k$ (when $2k/5<d<k/2$).

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arXiv2021

A better bound on the size of rainbow matchings

Aharoni and Howard conjectured that, for positive integers $n,k,t$ with $n\ge k$ and $n\ge t$, if $F_1,\ldots, F_t\subseteq {[n]\choose k}$ such that $|F_i|>{n\choose k}-{n-t+1\choose k}$ for $i\in [t]$ then there exist $e_i\in F_i$ for $i\in [t]$ such that $e_1,\ldots,e_t$ are pairwise disjoint. Huang, Loh, and Sudakov proved this conjecture for $t<n/(3k^2)$. In this paper, we show that this conjecture holds for $t\le n/(2k)$ and $n$ sufficiently large.

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arXiv2020

A characterization for graphs having strong parity factors

A graph $G$ has the \emph{strong parity property} if for every subset $X\subseteq V$ with $|X|$ even, $G$ has a spanning subgraph $F$ with minimum degree at least one such that $d_F(v)\equiv 1\pmod 2$ for all $v\in X$, $d_F(y)\equiv 0\pmod 2$ for all $y\in V(G)-X$. Bujtás, Jendrol and Tuza (On specific factors in graphs, \emph{Graphs and Combin.}, 36 (2020), 1391-1399.) introduced the concept and conjectured that every 2-edge-connected graph with minimum degree at least three has the strong parity property. In this paper, we give a characterization for graphs to have the strong parity property and construct a counterexample to disprove the conjecture proposed by Bujtás, Jendrol and Tuza.

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arXiv2020

A Degree Condition for Graphs Having All $(a,b)$-Parity Factors

Let $a$ and $b$ be positive integers such that $a\leq b$ and $a\equiv b\pmod 2$. We say that $G$ has all $(a, b)$-parity factors if $G$ has an $h$-factor for every function $h: V(G) \rightarrow \{a,a+2,\ldots,b-2,b\}$ with $b|V(G)|$ even and $h(v)\equiv b\pmod 2$ for all $v\in V(G)$. In this paper, we prove that every graph $G$ with $n\geq 3(b+1)(a+b)$ vertices has all $(a,b)$-parity factors if $δ(G)\geq (b^2-b)/a$, and for any two nonadjacent vertices $u,v \in V(G)$, $\max\{d_G(u),d_G(v)\}\geq \frac{bn}{a+b}$. Moreover, we show that this result is best possible in some sense.

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arXiv2020

Antifactors of regular bipartite graphs

Let $G=(X,Y;E)$ be a bipartite graph, where $X$ and $Y$ are color classes and $E$ is the set of edges of $G$. Lovász and Plummer \cite{LoPl86} asked whether one can decide in polynomial time that a given bipartite graph $G=(X,Y; E)$ admits a 1-anti-factor, that is subset $F$ of $E$ such that $d_F(v)=1$ for all $v\in X$ and $d_F(v)\neq 1$ for all $v\in Y$. Cornuéjols \cite{CHP} answered this question in the affirmative. Yu and Liu \cite{YL09} asked whether, for a given integer $k\geq 3$, every $k$-regular bipartite graph contains a 1-anti-factor. This paper answers this question in the affirmative.

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arXiv2020

Rainbow matchings for 3-uniform hypergraphs

Kühn, Osthus, and Treglown and, independently, Khan proved that if $H$ is a $3$-uniform hypergraph with $n$ vertices such that $n\in 3\mathbb{Z}$ and large, and $δ_1(H)>{n-1\choose 2}-{2n/3\choose 2}$, then $H$ contains a perfect matching. In this paper, we show that for $n\in 3\mathbb{Z}$ sufficiently large, if $F_1, \ldots, F_{n/3}$ are 3-uniform hypergrapghs with a common vertex set and $δ_1(F_i)>{n-1\choose 2}-{2n/3\choose 2}$ for $i\in [n/3]$, then $\{F_1,\dots, F_{n/3}\}$ admits a rainbow matching, i.e., a matching consisting of one edge from each $F_i$. This is done by converting the rainbow matching problem to a perfect matching problem in a special class of uniform hypergraphs.

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arXiv2020

An Ore-type Condition for Large $k$-factor and Disjoint Perfect Matchings

Win [\emph{J. Graph Theory} {\bf 6}(1982), 489--492] conjectured that a graph $G$ on $n$ vertices contains $k$ disjoint perfect matchings, if the degree sum of any two nonadjacent vertices is at least $n+k-2$, where $n$ is even and $n\geq k+2$. In this paper, we prove that Win's conjecture is true for $k\geq n/2$, where $n$ is sufficiently large. To show this result, we prove a theorem on $k$-factor in a graph under some Ore-type condition. Our main tools include Tutte's $k$-factor theorem, the Karush-Kuhn-Tucker theorem on convex optimization, and the solution to the longstanding 1-factor decomposition conjecture.

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arXiv2020

Regular graphs with equal matching number and independence number

Let $r\geq 3$ be an integer and $G$ be a graph. Let $δ(G), Δ(G)$, $α(G)$ and $μ(G)$ denotes minimum degree, maximum degree, independence number and matching number of $G$, respectively. Recently, Caro, Davila and Pepper proved $δ(G)α(G)\leq Δ(G)μ(G)$. Mohr and Rautenbach characterized the extremal graphs for non-regular graphs and 3-regular graphs. In this note, we characterize the extremal graphs for all $r$-regular graphs in term of Gallai-Edmonds Structure Theorem, which extends Mohr and Rautenbach's result.

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arXiv2019

Near perfect matchings in uniform hypergraphs

In this paper, we study degree conditions for the existence of large matchings in uniform hypergraphs. We prove that for integers $k,l,n$ with $k\ge 3$, $k/2 {n-l\choose k-l}-{(n-l)-(\lceil n/k \rceil-2)\choose 2}$, then $H$ has a matching covering all but a constant number of vertices. When $l=k-2$ and $k\ge 5$, such a matching is near perfect and our bound on $δ_l(H)$ is best possible. When $k=3$, with the help of an absorbing lemma of Hán, Person, and Schacht, our proof also implies that $H$ has a perfect matching, a result proved by K\" uhn, Osthus, and Treglown and, independently, of Kahn.

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arXiv2019

Fractional Factors, Component Factors and Isolated Vertex Conditions in Graphs

For a graph $G = (V, E)$, a {\em fractional $[a, b]$-factor} is a real valued function $h:E(G)\to [0,1]$ that satisfies $a \le ~ \sum_{e\in E_G(v)} h(e) ~ \le b$ for all $ v\in V(G)$, where $a$ and $b$ are real numbers and $E_G(v)$ denotes the set of edges incident with $v$. In this paper, we prove that the condition $\mathit{iso}(G-S) \le (k+\frac{1}{2})|S|$ is equivalent to the existence of fractional $[1,k+ \frac{1}{2}]$-factors, where ${\mathit{iso}}(G-S)$ denotes the number of isolated vertices in $G-S$. Using fractional factors as a tool, we construct component factors under the given isolated conditions. Namely, (i) a graph $G$ has a $\{P_2,C_3,P_5, \mathcal{T}(3)\}$-factor if and only if $\mathit{iso}(G-S) \le \frac{3}{2}|S|$ for all $S\subset V(G)$; (ii) a graph $G$ has a $\{K_{1,1}, K_{1,2}, \ldots,$ $K_{1,k}, \mathcal{T}(2k+1)\}$-factor ($k\ge 2$) if and only if $\mathit{iso}(G-S) \le (k+\frac{1}{2})|S|$ for all $S\subset V(G)$, where $\mathcal{T}(3)$ and $\mathcal{T}(2k+1)$ are two special families of trees.

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