Searcharxiv⌕ Search

arXiv subjects

Kaishun Wang

Publications and source records attributed to Kaishun Wang.

At least 37 records · Page 2Linked to original sources
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.

math.CO↗
arXiv2022

Cross $t$-intersecting families for symplectic polar spaces

Let $\mathscr{P}$ be a symplectic polar space over a finite field $\mathbb{F}_q$, and $\mathscr{P}_m$ denote the collection of all $k$-dimensional totally isotropic subspace in $\mathscr{P}$. Let $\mathscr{F}_1\subset\mathscr{P}_{m_1}$ and $\mathscr{F}_2\subset\mathscr{P}_{m_2}$ satisfy $\dim(F_1\cap F_2)\ge t$ for any $F_1\in\mathscr{F}_1$ and $F_2\in\mathscr{F}_2$. We say they are cross $t$-intersecting families. Moreover, we say they are trivial if each member of them contains a fixed $t$-dimensional totally isotropic subspace. In this paper, we show that cross $t$-intersecting families with maximum product of sizes are trivial. We also describe the structure of non-trivial $t$-intersecting families with maximum product of sizes.

math.CO↗
arXiv2022

Cross $t$-intersecting families for finite affine spaces

Denote the collection of all $k$-flats in $AG(n,\mathbb{F}_q)$ by $\mathscr{M}(k,n)$. Let $\mathscr{F}_1\subset\mathscr{M}(k_1,n)$ and $\mathscr{F}_2\subset\mathscr{M}(k_2,n)$ satisfy $\dim(F_1\cap F_2)\ge t$ for any $F_1\in\mathscr{F}_1$ and $F_2\in\mathscr{F}_2$. We say they are cross $t$-intersecting families. Moreover, we say they are trivial if each member of them contains a fixed $t$-flats in $AG(n,\mathbb{F}_q)$. In this paper, we show that cross $t$-intersecting families with maximum product of sizes are trivial. We also describe the structure of non-trivial $t$-intersecting families with maximum product of sizes.

math.CO↗
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.

math.CO↗
arXiv2021

Large non-trivial $t$-intersecting families for signed sets

For positive integers $n,r,k$ with $n\ge r$ and $k\ge2$, a set $\{(x_1,y_1),(x_2,y_2),\dots,(x_r,y_r)\}$ is called a $k$-signed $r$-set on $[n]$ if $x_1,\dots,x_r$ are distinct elements of $[n]$ and $y_1\dots,y_r\in[k]$. We say a $t$-intersecting family consisting of $k$-signed $r$-sets on $[n]$ is trivial if each member of this family contains a fixed $k$-signed $t$-set. In this paper, we determine the structure of large maximal non-trivial $t$-intersecting families. In particular, we characterize the non-trivial $t$-intersecting families with maximum size for $t\ge2$, extending a Hilton-Milner-type result for signed sets given by Borg.

math.CO↗
arXiv2021

Non-trivial $t$-intersecting families for the distance-regular graphs of bilinear forms

Let $V$ be an $(n+\ell)$-dimensional vector space over a finite field, and $W$ a fixed $\ell$-dimensional subspace of $V$. Write ${V\brack n,0}$ to be the set of all $n$-dimensional subspaces $U$ of $V$ satisfying $\dim(U\cap W)=0$. A family $\mathcal{F}\subseteq{V\brack n,0}$ is $t$-intersecting if $\dim(A\cap B)\geq t$ for all $A,B\in\mathcal{F}$. A $t$-intersecting family $\mathcal{F}\subseteq{V\brack n,0}$ is called non-trivial if $\dim(\cap_{F\in\mathcal{F}}F)<t$. In this paper, we describe the structure of non-trivial $t$-intersecting families of ${V\brack n,0}$ with large size. In particular, we show the structure of the non-trivial $t$-intersecting families with maximum size, which extends the Hilton-Milner Theorem for ${V\brack n,0}$.

math.CO↗
arXiv2020

On weak metric dimension of digraphs

Using the two way distance, we introduce the concepts of weak metric dimension of a strongly connected digraph $Γ$. We first establish lower and upper bounds for the number of arcs in $Γ$ by using the diameter and weak metric dimension of $Γ$, and characterize all digraphs attaining the lower or upper bound. Then we study a digraph with weak metric dimension $1$ and classify all vertex-transitive digraphs having weak metric dimension $1$. Finally, all digraphs of order $n$ with weak metric dimension $n-1$ or $n-2$ are determined.

math.CO↗
arXiv2020

Non-trivial $t$-intersecting families for symplectic polar spaces

Let $\mathscr{P}$ be a symplectic polar space over a finite field $\mathbb{F}_q$, and $\mathscr{P}_m$ denote the set of all $m$-dimensional subspaces in $\mathscr{P}$. We say a $t$-intersecting subfamily of $\mathscr{P}_m$ is trivial if there exists a $t$-dimensional subspace contained in each member of this family. In this paper, we determine the structure of maximum sized non-trivial $t$-intersecting subfamilies of $\mathscr{P}_m$.

math.CO↗
arXiv2020

Extremal even-cycle-free subgraphs of the complete transposition graphs

Given graphs $G$ and $H$, the generalized Turán number ${\rm ex}(G,H)$ is the maximum number of edges in an $H$-free subgraph of $G$. In this paper, we obtain an asymptotic upper bound on ${\rm ex}(CT_n,C_{2l})$ for any $n \ge 3$ and $l\geq2$, where $C_{2l}$ is the cycle of length $2l$ and $CT_n$ is the complete transposition graph which is defined as the Cayley graph on the symmetric group ${\rm S}_n$ with respect to the set of all transpositions of ${\rm S}_n$.

math.CO↗
arXiv2020

Perfect codes in Cayley sum graphs

A subset $C$ of the vertex set of a graph $Γ$ is called a perfect code of $Γ$ if every vertex of $Γ$ is at distance no more than one to exactly one vertex in $C$. Let $A$ be a finite abelian group and $T$ a square-free subset of $A$. The Cayley sum graph of $A$ with respect to the connection set $T$ is a simple graph with $A$ as its vertex set, and two vertices $x$ and $y$ are adjacent whenever $x+y\in T$. A subgroup of $A$ is said to be a subgroup perfect code of $A$ if the subgroup is a perfect code of some Cayley sum graph of $A$. In this paper, we give some necessary and sufficient conditions for a subset of $A$ to be a perfect code of a given Cayley sum graph of $A$. We also characterize all subgroup perfect codes of $A$.

math.CO↗
arXiv2020

Non-trivial $t$-intersecting families for vector spaces

Let $V$ be an $n$-dimensional vector space over a finite field $\mathbb{F}_q$. In this paper we describe the structure of maximal non-trivial $t$-intersecting families of $k$-dimensional subspaces of $V$ with large size. We also determine the non-trivial $t$-intersecting families with maximum size. In the special case when $t=1$ our result gives rise to the well-known Hilton-Milner Theorem for vector spaces.

math.CO↗
arXiv2020

Subgroup perfect codes in Cayley graphs

Let $Γ$ be a graph with vertex set $V(Γ)$. A subset $C$ of $V(Γ)$ is called a perfect code in $Γ$ if $C$ is an independent set of $Γ$ and every vertex in $V(Γ)\setminus C$ is adjacent to exactly one vertex in $C$. A subset $C$ of a group $G$ is called a perfect code of $G$ if there exists a Cayley graph of $G$ which admits $C$ as a perfect code. A group $G$ is said to be code-perfect if every proper subgroup of $G$ is a perfect code of $G$. In this paper we prove that a group is code-perfect if and only if it has no elements of order $4$. We also prove that a proper subgroup $H$ of an abelian group $G$ is a perfect code of $G$ if and only if the Sylow $2$-subgroup of $H$ is a perfect code of the Sylow $2$-subgroup of $G$. This reduces the problem of determining when a given subgroup of an abelian group is a perfect code to the case of abelian $2$-groups. Finally, we determine all subgroup perfect codes in any generalized quaternion group.

math.CO↗
arXiv2020

Non-trivial intersecting families for finite affine spaces

Guo and Xu determined the maximum size of intersecting families over finite affine spaces and showed that any family reaches maximum size must be trivial. In this paper, we characterize non-trivial intersecting family with maximum size.

math.CO↗
arXiv2020

Extremal $t$-intersecting families for direct products

In this paper, by shifting technique we study $t$-intersecting families for direct products where the ground set is divided into several parts. Assuming the size of each part is sufficiently large, we determine all extremal $t$-intersecting families for direct products. We also prove that every largest $t$-intersecting subfamily of a more general family introduced by Katona is trivial under certain conditions.

math.CO↗
arXiv2020

Thick weakly distance-regular digraphs

A weakly distance-regular digraph is thick if its attached scheme is regular. In this paper, we show that each commutative thick weakly distance-regular digraph has a thick weakly distance-regular subdigraph such that the corresponding quotient digraph falls into six families of thick weakly distance-regular digraphs up to isomorphism.

math.CO↗
arXiv2020

Intersecting families of vector spaces with maximum covering number

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_q$. Suppose that $\mathscr{F}$ is an intersecting family of $m$-dimensional subspaces of $V$. The covering number of $\mathscr{F}$ is the minimum dimension of a subspace of $V$ which intersects all elements of $\mathscr{F}$. In this paper, we give the tight upper bound for the size of $\mathscr{F}$ whose covering number is $m$, and describe the structure of $\mathscr{F}$ which reaches the upper bound. Moreover, we determine the structure of an maximum intersecting family of singular linear space with the maximum covering number.

math.CO↗