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Yongke Qu

Publications and source records attributed to Yongke Qu.

15 recordsLinked to original sources
arXiv2026

Extremal zero-sum sequences without elements of maximal order

Let $\mathsf D(G)$ denote the Davenport constant of a finite abelian group $G$, the maximal length of a minimal zero-sum sequence over $G$. Gao and Geroldinger conjectured that every minimal zero-sum sequence of length $\mathsf D(G)$~\cite[Conjecture~6.1]{GG99} and every zero-sumfree sequence of length $\mathsf D(G)-1$~\cite[Conjecture~4.1]{GG06} contains an element of order $\exp(G)$. We disprove both conjectures by proving a density-one theorem: for every fixed finite abelian group $H$ of exponent $M=2^ap^b$, where $a\ge2$, $b\ge1$, and $p$ is an odd prime, the set of integers $r\ge0$ for which $G_r=C_2^r\oplus H$ admits a minimal zero-sum sequence of length $\mathsf D(G_r)$ with no term of order $M$ has natural density one.

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arXiv2026

The Erdős--Ginzburg--Ziv theorem for finite groups

Let $G$ be a finite group, and let $p$ be the smallest prime divisor of $|G|$. The Gao constant $\mathsf{E}(G)$ is the least integer $\ell$ such that every sequence over $G$ of length $\ell$ contains a subsequence of length $|G|$ whose terms can be ordered to have product equal to the identity. In 1961, Erdős, Ginzburg and Ziv proved that every sequence of $2n-1$ integers has an $n$-term subsequence whose sum is divisible by $n$; equivalently, $\mathsf{E}(C_n)\leq 2n-1$, where $C_n$ denotes the cyclic group of order $n$. In this paper, we prove that $\mathsf{E}(G)\leq |G|+|G|/p-1$ whenever $G$ contains no cyclic subgroup of index $p$. Consequently, every finite non-cyclic group satisfies $\mathsf{E}(G)\leq |G|+|G|/p+p-2$, with equality if and only if $G$ contains a cyclic subgroup of index $p$. This establishes the conjectured upper bound for the Gao constant and provides a complete characterization of the equality case.

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arXiv2026

The Gao-Zhuang conjecture for the Heisenberg group over $\mathbb{F}_p$

Let $G$ be a finite nonabelian group. The small Davenport constant $\mathsf d(G)$ of $G$ is the largest integer $\ell$ such that there exists a product-one free sequence over $G$ of length $\ell$, while the Gao constant $E(G)$ of $G$ is the least integer $\ell$ such that every sequence over $G$ of length at least $\ell$ contains a product-one subsequence of length exactly $|G|$. A long-standing conjecture of Gao and Zhuang \cite{ZG2005} asserts that $E(G)=\mathsf d(G)+|G|$ for every finite nonabelian group $G$. Let $p$ be an odd prime and let $H_{p^3}=\operatorname{UT}_3(\mathbb F_p)$ be the finite Heisenberg group over $\mathbb F_p$. Godara and Sarkar proved the Gao-Zhuang equality for $H_{27}=\operatorname{UT}_3(\mathbb F_3)$ and asked whether the same equality holds for $H_{p^3}$ for every odd prime $p$. Recently, Volkmann proved that $\mathsf d(H_{p^3})=3p-3$. In this paper, we determine the Gao constant of $H_{p^3}$ and prove that $E(H_{p^3})=\mathsf d(H_{p^3})+|H_{p^3}|=p^3+3p-3$. Together with the known abelian and cyclic-index cases, this completes the verification of the Gao-Zhuang equality for all groups of order $p^3$, for every prime $p$.

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arXiv2026

The equality between the Erdős-Ginzburg-Ziv constant and the short product-one constant for finite nonabelian groups

Let $G$ be a finite group, and let $\exp(G)$ denote its exponent. The Erdős-Ginzburg-Ziv constant $s(G)$ is the least integer forcing a product-one subsequence of length $\exp(G)$, while the short product-one constant $η(G)$ is the least integer forcing a nonempty product-one subsequence of length at most $\exp(G)$. The natural nonabelian extension of a conjecture [W. Gao, \emph{On zero-sum subsequences of restricted size II}, Discrete Math. 2003] on the Erdős-Ginzburg-Ziv constant in finite abelian groups predicts that $s(G)=η(G)+\exp(G)-1.$ We confirm this equality for every finite nonabelian group $G$ having a cyclic subgroup of index $p$, where $p$ is the smallest prime divisor of $|G|$. As further consequences, we determine all generalized Erdős-Ginzburg-Ziv constants $s_{m\exp(G)}(G)$ for this family of groups.

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arXiv2021

additive bases of abelian groups of rank 2

Let $G$ be a finite abelian group and $p$ be the smallest prime dividing $|G|$. Let $S$ be a sequence over $G$. We say that $S$ is regular if for every proper subgroup $H \subsetneq G$, $S$ contains at most $|H|-1$ terms from $H$. Let $\mathsf c_0(G)$ be the smallest integer $t$ such that every regular sequence $S$ over $G$ of length $|S|\geq t$ forms an additive basis of $G$, i.e., $\sum(S)=G$. The invariant $\mathsf c_0(G)$ was first studied by Olson and Peng in 1980's, and since then it has been determined for all finite abelian groups except for the groups with rank 2 and a few groups of rank 3 or 4 with order less than $10^8$. In this paper, we focus on the remaining case concerning groups of rank 2. It was conjectured by Gao et al. (Acta Arith. 168 (2015) 247-267) that $\mathsf c_0(G)=m(G)$. We confirm the conjecture for the case when $G=C_{n_1}\oplus C_{n_2}$ with $n_1|n_2$, $n_1\geq 2p$, $p\geq 3$ and $n_1n_2\geq 72p^6$.

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arXiv2021

Extremal product-one free sequences and $|G|$-product-one free sequences of a metacyclic group

Let $G$ be a multiplicatively written finite group. We denote by $\mathsf E(G)$ the smallest integer $t$ such that every sequence of $t$ elements in $G$ contains a product-one subsequence of length $|G|$. In 1961, Erdős, Ginzburg and Ziv proved that $\mathsf E(G)\leq 2|G|-1$ for every finite abelian group $G$ and this result is known as the Erdős-Ginzburg-Ziv Theorem. In 2005, Zhuang and Gao conjectured that $\mathsf E(G)=\mathsf d(G)+|G|$ for every finite group, where $\mathsf d(G)$ is the small Davenport constant. Very recently, we confirmed this conjecture for the case when $G=\langle x, y| x^p=y^m=1, x^{-1}yx=y^r\rangle$ where $p$ is the smallest prime divisor of $|G|$ and $\mbox{gcd}(p(r-1), m)=1$. In this paper, we study the associated inverse problems on $\mathsf d(G)$ and $\mathsf E(G)$. Our main results characterize the structure of any product-one free sequence with extremal length $\mathsf d(G)$, and that of any $|G|$-product-one free sequence with extremal length $\mathsf E(G)-1$.

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arXiv2021

On the invariant E(G) for groups of odd order

Let $G$ be a multiplicatively written finite group. We denote by $\mathsf E(G)$ the smallest integer $t$ such that every sequence of $t$ elements in $G$ contains a product-one subsequence of length $|G|$. In 1961, Erdős, Ginzburg and Ziv proved that $\mathsf E(G)\leq 2|G|-1$ for every finite solvable group $G$ and this result is well known as the Erdős-Ginzburg-Ziv Theorem. In 2010, Gao and Li improved this result to $\mathsf E(G)\leq\frac{7|G|}{4}-1$ and they conjectured that $\mathsf E(G)\leq \frac{3|G|}{2}$ holds for any finite non-cyclic group. In this paper, we confirm the conjecture for all finite non-cyclic groups of odd order.

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arXiv2021

On a conjecture of Zhuang and Gao

Let $G$ be a multiplicatively written finite group. We denote by $\mathsf E(G)$ the smallest integer $t$ such that every sequence of $t$ elements in $G$ contains a product-one subsequence of length $|G|$. In 1961, Erdős, Ginzburg and Ziv proved that $\mathsf E(G)\leq 2|G|-1$ for every finite ablian group $G$ and this result is known as the Erdős-Ginzburg-Ziv Theorem. In 2005, Zhuang and Gao conjectured that $\mathsf E(G)=\mathsf d(G)+|G|$, where $\mathsf d(G)$ is the small Davenport constant. In this paper, we confirm the conjecture for the case when $G=\langle x, y| x^p=y^m=1, x^{-1}yx=y^r\rangle$, where $p$ is the smallest prime divisor of $|G|$ and $\mbox{gcd}(p(r-1), m)=1$.

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arXiv2021

Additive bases of $C_3\oplus C_{3q}$

Let $G$ be a finite abelian group and $p$ be the smallest prime dividing $|G|$. Let $S$ be a sequence over $G$. We say that $S$ is regular if for every proper subgroup $H \subsetneq G$, $S$ contains at most $|H|-1$ terms from $H$. Let $\mathsf c_0(G)$ be the smallest integer $t$ such that every regular sequence $S$ over $G$ of length $|S|\geq t$ forms an additive basis of $G$, i.e., $\sum(S)=G$. The invariant $\mathsf c_0(G)$ was first studied by Olson and Peng in 1980's, and since then it has been determined for all finite abelian groups except for the groups with rank 2 and a few groups of rank 3 or 4 with order less than $10^8$. In this paper, we focus on the remaining case concerning groups of rank 2. It was conjectured by the first author and Han (Int. J. Number Theory 13 (2017) 2453-2459) that $\mathsf c_0(G)=pn+2p-3$ where $G=C_p\oplus C_{pn}$ with $n\geq 3$. We confirm the conjecture for the case when $p=3$ and $n=q \,(\geq 5)$ is a prime number.

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arXiv2015

A problem of Wang on Davenport constant for the multiplicative semigroup of the quotient ring of $\F_2[x]$

Let $\F_q[x]$ be the ring of polynomials over the finite field $\F_q$, and let $f$ be a polynomial of $\F_q[x]$. Let $R=\frac{\F_q[x]}{(f)}$ be a quotient ring of $\F_q[x]$ with $0\neq R\neq \F_q[x]$. Let $\mathcal{S}_R$ be the multiplicative semigroup of the ring $R$, and let ${\rm U}(\mathcal{S}_R)$ be the group of units of $\mathcal{S}_R$. The Davenport constant ${\rm D}(\mathcal{S}_R)$ of the multiplicative semigroup $\mathcal{S}_R$ is the least positive integer $\ell$ such that for any $\ell$ polynomials $g_1,g_2,\ldots,g_{\ell}\in \F_q[x]$, there exists a subset $I\subsetneq [1,\ell]$ with $$\prod\limits_{i\in I} g_i \equiv \prod\limits_{i=1}^{\ell} g_i\pmod f.$$ In this manuscript, we proved that for the case of $q=2$, $${\rm D}({\rm U}(\mathcal{S}_R))\leq {\rm D}(\mathcal{S}_R)\leq {\rm D}({\rm U}(\mathcal{S}_R))+δ_f,$$ where \begin{displaymath} δ_f=\left\{\begin{array}{ll} 0 & \textrm{if $\gcd(x*(x+1_{\mathbb{F}_2}),\ f)=1_{\F_{2}}$}\\ 1 & \textrm{if $\gcd(x*(x+1_{\mathbb{F}_2}),\ f)\in \{x, \ x+1_{\mathbb{F}_2}\}$}\\ 2 & \textrm{if $gcd(x*(x+1_{\mathbb{F}_2}),f)=x*(x+1_{\mathbb{F}_2}) $}\\ \end{array} \right. \end{displaymath} which partially answered an open problem of Wang on Davenport constant for the multiplicative semigroup of $\frac{\F_q[x]}{(f)}$ (G.Q. Wang, \emph{Davenport constant for semigroups II,} Journal of Number Theory, 155 (2015) 124--134).

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arXiv2014

Davenport constant of the multiplicative semigroup of the quotient ring $\frac{\F_p[x]}{\langle f(x)\rangle}$

Let $\mathcal{S}$ be a finite commutative semigroup. The Davenport constant of $\mathcal{S}$, denoted $D(\mathcal{S})$, is defined to be the least positive integer $d$ such that every sequence $T$ of elements in $\mathcal{S}$ of length at least $d$ contains a subsequence $T'$ with the sum of all terms from $T'$ equaling the sum of all terms from $T$. Let $\F_p[x]$ be a polynomial ring in one variable over the prime field $\F_p$, and let $f(x)\in \F_p[x]$. In this paper, we made a study of the Davenport constant of the multiplicative semigroup of the quotient ring $\frac{\F_p[x]}{\langle f(x)\rangle}$. Among other results, we mainly prove that, for any prime $p>2$ and any polynomial $f(x)\in \F_p[x]$ which can be factorized into several pairwise non-associted irreducible polynomials in $\F_p[x]$, then $$D(\mathcal{S}_{f(x)}^p)=D(U(\mathcal{S}_{f(x)}^p)),$$ where $\mathcal{S}_{f(x)}^p$ denotes the multiplicative semigroup of the quotient ring $\frac{\F_p[x]}{\langle f(x)\rangle}$ and $U(\mathcal{S}_{f(x)}^p)$ denotes the group of units of the semigroup $\mathcal{S}_{f(x)}^p$.

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arXiv2014

On reformulated zagreb indices with respect to tricyclic graphs

The authors Mili$\breve{c}$evi$\acute{c}$ et al. introduced the reformulated Zagreb indices, which is a generalization of classical Zagreb indices of chemical graph theory. In the paper, we characterize the extremal properties of the first reformulated Zagreb index. We first introduce some graph operations which increase or decrease this index. Furthermore, we will determine the extremal tricyclic graphs with minimum and maximum the first Zagreb index by these graph operations.

math.CO↗
arXiv2012

On The Critical Number of Finite Groups (II)

Let G be a finite group and S a subset of G\{0}. We call S an additive basis of G if every element of G can be expressed as a sum over a nonempty subset in some order. Let cr(G) be the smallest integer t such that every subset of G\{0} of cardinality t is an additive basis of G. In this paper, we determine cr(G) for the following cases: (i) G is a finite nilpotent group; (ii) G is a group of even order which possesses a subgroup of index 2.

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arXiv2011

Extremal incomplete sets in finite abelian groups

Let $G$ be a finite abelian group. The critical number ${\rm cr}(G)$ of $G$ is the least positive integer $\ell$ such that every subset $A\subseteq G\setminus\{0\}$ of cardinality at least $\ell$ spans $G$, i.e., every element of $G$ can be written as a nonempty sum of distinct elements of $A$. The exact values of the critical number have been completely determined recently for all finite abelian groups. The structure of these sets of cardinality ${\rm cr}(G)-1$ which fail to span $G$ has also been characterized except for the case that $|G|$ is an even number and the case that $|G|=pq$ with $p,q$ are primes. In this paper, we characterize these extremal subsets for $|G|\geq 36$ is an even number, or $|G|=pq$ with $p,q$ are primes and $q\geq 2p+3$.

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arXiv2011

On the number of subsequences with a given sum in a finite abelian group

Suppose $G$ is a finite abelian group and $S$ is a sequence of elements in $G$. For any element $g$ of $G$, let $N_g(S)$ denote the number of subsequences of $S$ with sum $g$. The purpose of this paper is to investigate the lower bound for $N_g(S)$. In particular, we prove that either $N_g(S)=0$ or $N_g(S) \ge 2^{|S|-D(G)+1}$, where $D(G)$ is the smallest positive integer $\ell$ such that every sequence over $G$ of length at least $\ell$ has a nonempty zero-sum subsequence. We also characterize the structures of the extremal sequences for which the equality holds for some groups.

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