arXiv subjects

Mei Lu

Publications and source records attributed to Mei Lu.

At least 55 records · Page 3Linked to original sources

arXiv2024

Turán number of complete bipartite graphs with bounded matching number

Let $\mathscr{F}$ be a family of graphs. A graph $G$ is $\mathscr{F}$-free if $G$ does not contain any $F\in \mathcal{F}$ as a subgraph. The Turán number $ex(n, \mathscr{F})$ is the maximum number of edges in an $n$-vertex $\mathscr{F}$-free graph. Let $M_{s}$ be the matching consisting of $ s $ independent edges. Recently, Alon and Frank determined the exact value of $ex(n,\{K_{m},M_{s+1}\})$. Gerbner obtained several results about $ex(n,\{F,M_{s+1}\})$ when $F$ satisfies certain proportions. In this paper, we determine the exact value of $ex(n,\{K_{l,t},M_{s+1}\})$ when $s, n$ are large enough for every $3\leq l\leq t$. When $n$ is large enough, we also show that $ex(n,\{K_{2,2}, M_{s+1}\})=n+{s\choose 2}-\left\lceil\frac{s}{2}\right\rceil$ for $s\ge 12$ and $ex(n,\{K_{2,t},M_{s+1}\})=n+(t-1){s\choose 2}-\left\lceil\frac{s}{2}\right\rceil$ when $t\ge 3$ and $s$ is large enough.

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arXiv2024

Hilton-Milner theorem for $k$-multisets

Let $ k, n \in \mathbb{N}^+ $ and $ m \in \mathbb{N}^+ \cup \{\infty \} $. A $ k $-multiset in $ [n]_m $ is a $ k $-set whose elements are integers from $ \{1, 2, \ldots, n\} $, and each element is allowed to have at most $ m $ repetitions. A family of $ k $-multisets in $ [n]_m $ is said to be intersecting if every pair of $ k $-multisets from the family have non-empty intersection. In this paper, we give the size and structure of the largest non-trivial intersecting family of $ k $-multisets in $ [n]_m $ for $ n \geq k + \lceil k/m \rceil $. In the special case when $m=\infty$, our result gives rise to an unbounded multiset version for Hilton-Milner Theorem given by Meagher and Purdy. Furthermore, our main theorem unites the statements of the Hilton-Milner Theorem for finite sets and unbounded multisets.

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arXiv2024

The generalized Tur'{a}n number of long cycles in graphs and bipartite graphs

Given a graph $T$ and a family of graphs $\mathcal{F}$, the maximum number of copies of $T$ in an $\mathcal{F}$-free graph on $n$ vertices is called the generalized Turán number, denoted by $ex(n, T , \mathcal{F})$. When $T= K_2$, it reduces to the classical Turán number $ex(n, \mathcal{F})$. Let $ex_{bip}(b,n, T , \mathcal{F})$ be the maximum number of copies of $T$ in an $\mathcal{F}$-free bipartite graph with two parts of sizes $b$ and $n$, respectively. Let $P_k$ be the path on $k$ vertices, $\mathcal{C}_{\ge k}$ be the family of all cycles with length at least $k$ and $M_k$ be a matching with $k$ edges. In this article, we determine $ex_{bip}(b,n, K_{s,t}, \mathcal{C}_{\ge 2n-2k})$ exactly in a connected bipartite graph $G$ with minimum degree $δ(G) \geq r\ge 1$, for $b\ge n\ge 2k+2r$ and $k\in \mathbb{Z}$, which generalizes a theorem of Moon and Moser, a theorem of Jackson and gives an affirmative evidence supporting a conjecture of Adamus and Adamus. As corollaries of our main result, we determine $ex_{bip}(b,n, K_{s,t}, P_{2n-2k})$ and $ex_{bip}(b,n, K_{s,t}, M_{n-k})$ exactly in a connected bipartite graph $G$ with minimum degree $δ(G) \geq r\ge 1$, which generalizes a theorem of Wang. Moreover, we determine $ex(n, K_{s,t}, \mathcal{C}_{\ge k})$ and $ex(n, K_{s,t}, P_{k})$ respectively in a connected graph $G$ with minimum degree $δ(G) \geq r\ge 1$, which generalizes a theorem of Lu, Yuan and Zhang.

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arXiv2024

The inversion number of dijoins and blow-up digraphs

For an oriented graph $D$, the $inversion$ of $X \subseteq V(D)$ in $D$ is the digraph obtained from $D$ by reversing the direction of all arcs with both ends in $X$. The inversion number of $D$, denoted by $inv(D)$, is the minimum number of inversions needed to transform $D$ into an acyclic digraph. In this paper, we first show that $inv (\overrightarrow{C_3} \Rightarrow D)= inv(D) +1$ for any oriented graph $\textit{D}$ with even inversion number $inv(D)$, where the dijoin $\overrightarrow{C_3} \Rightarrow D$ is the oriented graph obtained from the disjoint union of $\overrightarrow{C_3}$ and $D$ by adding all arcs from $\overrightarrow{C_3}$ to $D$. Thus we disprove the conjecture of Aubian el at. \cite{2212.09188} and the conjecture of Alon el at. \cite{2212.11969}. We also study the blow-up graph which is an oriented graph obtained from a tournament by replacing all vertices into oriented graphs. We construct a tournament $T$ with order $n$ and $inv(T)=\frac{n}{3}+1$ using blow-up graphs.

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arXiv2024

A polynomial time algorithm to find star chromatic index on bounded treewidth graphs with given maximum degree

A star edge coloring of a graph $G$ is a proper edge coloring with no 2-colored path or cycle of length four. The star edge coloring problem is to find an edge coloring of a given graph $G$ with minimum number $k$ of colors such that $G$ admits a star edge coloring with $k$ colors. This problem is known to be NP-complete. In this paper, for a bounded treewidth graph with given maximum degree, we show that it can be solved in polynomial time.

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arXiv2023

Erdős-Ko-Rado Theorem for Bounded Multisets

Let $ k, m, n $ be positive integers with $ k \geq 2 $. A $ k $-multiset of $ [n]_m $ is a collection of $ k $ integers from the set $ \{1, 2, \ldots, n\} $ in which the integers can appear more than once but at most $ m $ times. A family of such $ k $-multisets is called an intersecting family if every pair of $ k $-multisets from the family have non-empty intersection. A finite sequence of real numbers $\{a_1,a_2,\ldots,a_n\}$ is said to be unimodal if there is some $k\in \{1,2,\ldots,n\}$, such that $a_1\leq a_2\leq\ldots\leq a_{k-1}\leq a_k\geq a_{k+1}\geq \ldots\geq a_n$. Given $m,n,k$, denote $C_{k,l}$ as the coefficient of $x^k$ in the generating function $(\sum_{i=1}^mx^i)^l$, where $1\leq l\leq n$. In this paper, we first show that the sequence of $\{C_{k,1},C_{k,2},\ldots,C_{k,n}\}$ is unimodal. Then we use this as a tool to prove that the intersecting family in which every $ k $-multiset contains a fixed element attains the maximum cardinality for $ n \geq k + \lceil k/m\rceil $. In the special case when $m = 1$ and $m=\infty$, our result gives rise to the famous Erdős-Ko-Rado Theorem and an unbounded multiset version for this problem given by Meagher and Purdy, respectively. The main result in this paper can be viewed as a bounded multiset version of the Erdős-Ko-Rado Theorem.

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arXiv2022

Triangles in r-wise t-intersecting families

Let $t$, $r$, $k$ and $n$ be positive integers and $\mathcal{F}$ a family of $k$-subsets of an $n$-set $V$. The family $ \CF $ is $ r $-wise $ t $-intersecting if for any $ F_1, \ldots, F_r \in \CF $, we have $ \abs{\cap_{i = 1}^{r}F_i}\gs t $. An $ r $-wise $ t $-intersecting family of $ r + 1 $ sets $ \{T_1, \ldots, T_{r + 1}\} $ is called an $ (r + 1,t) $-triangle if $ |T_1 \cap \cdots \cap T_{r + 1}| \ls t - 1 $. In this paper, we prove that if $ n \gs n_0(r, t, k) $, then the $ r $-wise $ t $-intersecting family $ \CF \subseteq \binom{[n]}{k} $ containing the most $ (r + 1,t) $-triangles is isomorphic to $ \curlybraces{F \in \binom{[n]}{k}: \abs{F \cap [r + t]} \gs r + t - 1} $. This can also be regarded as a generalized Turán type result.

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arXiv2022

Edge pancyclic derangement graphs

We consider the derangement graph in which the vertices are permutations of $\{ 1,\ldots, n\}$. Two vertices are joined by an edge if the corresponding permutations differ in every position. The derangement graph is known to be Hamiltonian and Hamilton-connected. In this note, we show that the derangement graph is edge pancyclic if $n\ge 4$.

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arXiv2022

Some intersection theorems for finite sets

Let $n$, $r$, $k_1,\ldots,k_r$ and $t$ be positive integers with $r\geq 2$, and $\mathcal{F}_i\ (1\leq i\leq r)$ a family of $k_i$-subsets of an $n$-set $V$. The families $\mathcal{F}_1,\ \mathcal{F}_2,\ldots,\mathcal{F}_r$ are said to be $r$-cross $t$-intersecting if $|F_1\cap F_2\cap\cdots\cap F_r|\geq t$ for all $F_i\in\mathcal{F}_i\ (1\leq i\leq r),$ and said to be non-trivial if $|\cap_{1\leq i\leq r}\cap_{F\in\mathcal{F}_i}F|<t$. If the $r$-cross $t$-intersecting families $\mathcal{F}_1,\ldots,\mathcal{F}_r$ satisfy $\mathcal{F}_1=\cdots=\mathcal{F}_r=\mathcal{F}$, then $\mathcal{F}$ is well known as $r$-wise $t$-intersecting family. In this paper, we describe the structure of non-trivial $r$-wise $t$-intersecting families with maximum size, and give a stability result for these families. We also determine the structure of non-trivial $2$-cross $t$-intersecting families with maximum product of their sizes.

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arXiv2022

On the $ P_3 $-hull numbers of $ q $-Kneser graphs and Grassmann graphs

Let $S$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_q$, where $q$ is necessarily a prime power. Denote $K_q(n,k)$ (resp. $J_q(n,k)$) to be the \emph{$q$-Kneser graph} (resp. \emph{Grassmann graph}) for $k\geq 1$ whose vertices are the $k$-dimensional subspaces of $S$ and two vertices $v_1$ and $v_2$ are adjacent if $\dim(v_1\cap v_2)=0$ (resp. $\dim(v_1\cap v_2)=k-1$). We consider the infection spreading in the $ q $-Kneser graphs and the Grassmann graphs: a vertex gets infected if it has at least two infected neighbors. In this paper, we compute the $ P_3 $-hull numbers of $K_q(n,k)$ and $J_q(n,k)$ respectively, which is the minimum size of a vertex set that eventually infects the whole graph.

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arXiv2022

$r$-cross $t$-intersecting families for vector spaces

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_q$, and ${V\brack k}$ denote the family of all $k$-dimensional subspaces of $V$. The families $\mathcal{F}_1\subseteq{V\brack k_1},\mathcal{F}_2\subseteq{V\brack k_2},\ldots,\mathcal{F}_r\subseteq{V\brack k_r}$ are said to be $r$-cross $t$-intersecting if $\dim(F_1\cap F_2\cap\cdots\cap F_r)\geq t$ for all $F_i\in\mathcal{F}_i,\ 1\leq i\leq r.$ The $r$-cross $t$-intersecting families $\mathcal{F}_1$, $\mathcal{F}_2,\ldots,\mathcal{F}_r$ are said to be non-trivial if $\dim(\cap_{1\leq i\leq r}\cap_{F\in\mathcal{F}_i}F)<t$. In this paper, we first determine the structure of $r$-cross $t$-intersecting families with maximum product of their sizes. As a consequence, we partially prove one of Frankl and Tokushige's conjectures about $r$-cross $1$-intersecting families for vector spaces. Then we describe the structure of non-trivial $r$-cross $t$-intersecting families $\mathcal{F}_1$, $\mathcal{F}_2,\ldots,\mathcal{F}_r$ with maximum product of their sizes under the assumptions $r=2$ and $\mathcal{F}_1=\mathcal{F}_2=\cdots=\mathcal{F}_r=\mathcal{F}$, respectively, where the $\mathcal{F}$ in the latter assumption is well known as $r$-wise $t$-intersecting family. Meanwhile, stability results for non-trivial $r$-wise $t$-intersecting families are also been proved.

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arXiv2021

The treewidth of 2-section of hypergraphs

Let $H=(V,F)$ be a simple hypergraph without loops. $H$ is called linear if $|f\cap g|\le 1$ for any $f,g\in F$ with $f\not=g$. The $2$-section of $H$, denoted by $[H]_2$, is a graph with $V([H]_2)=V$ and for any $ u,v\in V([H]_2)$, $uv\in E([H]_2)$ if and only if there is $ f\in F$ such that $u,v\in f$. The treewidth of a graph is an important invariant in structural and algorithmic graph theory. In this paper, we consider the treewidth of the $2$-section of a linear hypergraph. We will use the minimum degree, maximum degree, anti-rank and average rank of a linear hypergraph to determine the upper and lower bounds of the treewidth of its $2$-section. Since for any graph $G$, there is a linear hypergraph $H$ such that $[H]_2\cong G$, we provide a method to estimate the bound of treewidth of graph by the parameters of the hypergraph.

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arXiv2021

On graphs with exactly one anti-adjacency eigenvalue and beyond

The anti-adjacency matrix of a graph is constructed from the distance matrix of a graph by keeping each row and each column only the largest distances. This matrix can be interpreted as the opposite of the adjacency matrix, which is instead constructed from the distance matrix of a graph by keeping in each row and each column only the distances equal to 1. The (anti-)adjacency eigenvalues of a graph are those of its (anti-)adjacency matrix. Employing a novel technique introduced by Haemers [Spectral characterization of mixed extensions of small graphs, Discrete Math. 342 (2019) 2760--2764], we characterize all connected graphs with exactly one positive anti-adjacency eigenvalue, which is an analog of Smith's classical result that a connected graph with exactly one positive adjacency eigenvalue iff it is a complete multipartite graph. On this basis, we identify the connected graphs with all but at most two anti-adjacency eigenvalues equal to $-2$ and $0$. Moreover, for the anti-adjacency matrix we determine the HL-index of graphs with exactly one positive anti-adjacency eigenvalue, where the HL-index measures how large in absolute value may be the median eigenvalues of a graph. We finally propose some problems for further study.

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arXiv2021

Anti-Ramsey problems in the generalized Petersen graphs for cycles

The anti-Ramsey number $Ar(G,H)$ is the maximum number of colors in an edge-coloring of $G$ with no rainbow copy of $H$. In this paper, we determine the exact anti-Ramsey number in the generalized Petersen graph $P_{n,k}$ for cycles $C_d$, where $1\leq k\leq \lfloor \frac{n-1}{2} \rfloor$ and $5\le d \le 6$. We also give an algorithm to obtain the upper bound or lower bound of anti-Ramsey number.

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arXiv2021

Treewidth of the generalized Kneser graphs

Let $n$, $k$ and $t$ be integers with $1\leq t< k \leq n$. The \emph{generalized Kneser graph} $K(n,k,t)$ is a graph whose vertices are the $k$-subsets of a fixed $n$-set, where two $k$-subsets $A$ and $B$ are adjacent if $|A\cap B|<t$. The graph $K(n,k,1)$ is the well-known \emph{Kneser graph}. In 2014, Harvey and Wood determined the exact treewidth of the Kneser graphs for large $n$ with respect to $k$. In this paper, we give the exact treewidth of the generalized Kneser graphs for $t\geq2$ and large $n$ with respect to $k$ and $t$. In the special case when $t=k-1$, the graph $K(n,k,k-1)$ usually denoted by $\overline{J(n,k)}$ which is the complement of the Johnson graph $J(n,k)$. We give a more precise result for the exact value of the treewidth of $\overline{J(n,k)}$ for any $n$ and $k$.

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arXiv2021

Rainbow Independent Sets in Cycles

For a given class ${\cal C}$ of graphs and given integers $m \le n$, let $f_{\cal C}(n,m)$ be the minimal number $k$ such that every $k$ independent $n$-sets in any graph belonging to ${\cal C}$ have a (possibly partial) rainbow independent $m$-set. In this paper, we consider the case ${\cal C}=\{C_{2s+1}\}$ and show that $f_{C_{2s+1}}(s, s) = s$. Our result is a special case of the conjecture (Conjecture 2.9) proposed by Aharoni et al in \cite{Aharoni}.

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arXiv2021

Treewidth of the $q$-Kneser graphs

Let $V$ be an $n$-dimensional vector space over a finite field $\mathbb{F}_q$, where $q$ is a prime power. Define the \emph{generalized $q$-Kneser graph} $K_q(n,k,t)$ to be the graph whose vertices are the $k$-dimensional subspaces of $V$ and two vertices $F_1$ and $F_2$ are adjacent if $\dim(F_1\cap F_2)<t$. Then $K_q(n,k,1)$ is the well-known $q$-Kneser graph. In this paper, we determine the treewidth of $K_q(n,k,t)$ for $n\geq 2t(k-t+1)+k+1$ and $t\ge 1$ exactly. Note that $K_q(n,k,k-1)$ is the complement of the Grassmann graph $G_q(n,k)$. We give a more precise result for the treewidth of $\overline{G_q(n,k)}$ for any possible $n$, $k$ and $q$.

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arXiv2021

$w$-Dominating Set Problem on Graphs of Bounded Treewidth

Let $G=(V,E)$ be a graph. Let $w$ be a positive integer. A $w$-dominating set is a vertex subset $S$ such that for all $v\in V$, either $v\in S$ or it has at least $w$ neighbors in $S$. The $w$-Dominating Set problem is to find the minimum $w$-dominating set. The $L$-Max $w$-Dominating Set problem is to find the vertex subset $S$ of cardinality at most $L$ that maximizes $|S|+|\{v\in V\setminus S~|~|N(v)\cap S|\geq w\}|$, where $N(v)=\{u|uv\in E\}$. In this paper, we give polynomial time algorithms to $w$-Dominating Set problem and $L$-Max $w$-Dominating Set problem on graphs of bounded treewidth.

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