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Tony W. H. Wong

Publications and source records attributed to Tony W. H. Wong.

At least 19 recordsLinked to original sources
arXiv2026

Sharp Nordhaus-Gaddum bounds for throttling

Throttling is a graph optimization problem, where the throttling number of a graph is the minimum sum or minimum product of the number of vertices in an initial set and the time required to complete a certain graph operation. A Nordhaus-Gaddum bound refers to an upper or lower bound of the sum or product of a graph parameter together with that of its complement. In this paper, we study the Nordhaus-Gaddum sum and product bounds of the various throttling numbers (sum throttling and product throttling with or without initial cost). Graph operations considered are standard zero forcing, positive semidefinite forcing, power domination, and Cops and Robbers.

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arXiv2026

On Zeckendorf-Niven numbers and arithmetic progressions

A positive integer is Zeckendorf-Niven (respectively, Lucas-Niven) if it is divisible by the number of summands in its Zeckendorf decomposition (respectively, Lucas decomposition). We show that there exist infinitely many Zeckendorf-Niven numbers and Lucas-Niven numbers in every arithmetic progression. Furthermore, we provide bounds on the maximum number of consecutive Zeckendorf-Niven terms in certain arithmetic progressions.

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arXiv2026

Atoms in the Semigroup of Non-Negative Integer Matrices

In the semigroup $M_2(\mathbb{N}_0)^\bullet$, two-by-two matrices with non-negative integer entries and non-zero determinant, we study the factorization of matrices into atoms, or irreducible matrices. In 2022, Baeth et al. listed some fundamental classes of atoms in $M_2(\mathbb{N}_0)^\bullet$; however, the factorability of most matrices in $M_2(\mathbb{N}_0)^\bullet$ remains unknown. We identify two additional classes of atoms: a class of atoms with determinant $p$, $2p$, or $4p$, for $p$ prime, and a class of atoms in which the main diagonal is much "larger" than the off-diagonal (or vice versa). Finally, we show that bisymmetric matrices with relatively prime entries are a divisor-closed subset of $M_2(\mathbb{N}_0)^\bullet$ and use a factor search algorithm to classify bisymmetric atoms of $M_2(\mathbb{N}_0)^\bullet$ with minimum entry up to 4000.

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arXiv2025

On Sierpiński and Riesel Repdigits and Repintegers

For positive integers $b\geq 2$, $k<b$, and $t$, we say that an integer $k_b^{(t)}$ is a $b$-repdigit if $k_b^{(t)}$ can be expressed as the digit $k$ repeated $t$ times in base-$b$ representation, i.e., $k_b^{(t)} =k(b^t-1)/(b-1)$. In the case of $k=1$, we say that $1_b^{(t)}$ is a $b$-repunit. In this article, we investigate the existsence of $b$-repdigits and $b$-repunits among the sets of Sierpiński numbers and Riesel numbers. A Sierpiński number is defined as an odd integer $k$ for which $k\cdot 2^n+1$ is composite for all positive integers $n$ and Riesel numbers are similarly defined for the expression $k\cdot 2^n-1$.

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arXiv2025

A further investigation on covering systems with odd moduli

Erdős first introduced the idea of covering systems in 1950. Since then, much of the work in this area has concentrated on identifying covering systems that meet specific conditions on their moduli. Among the central open problems in this field is the well-known odd covering problem. In this paper, we investigate a variant of that problem, where one odd integer is permitted to appear multiple times as a modulus in the covering system, while all remaining moduli are distinct odd integers greater than 1.

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arXiv2025

Sharp bounds for product and sum throttling numbers

Throttling in graphs optimizes a sum or product of resources used, such as the number of vertices in an initial set, and time required, such as the propagation time, to complete a given task. We introduce a new technique to establish sharp upper bounds in terms of graph order for sum throttling and initial cost product throttling for power domination. Furthermore, we establish sharp bounds on possible changes of the product throttling number, both with and without initial cost, caused by certain graph operations for standard zero forcing, positive semidefinite forcing, and power domination.

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arXiv2024

Paired $(n-1)$-to-$(n-1)$ disjoint path covers in bipartite transposition-like graphs

A paired $k$-to-$k$ disjoint path cover of a graph $G$ is a collection of pairwise disjoint path subgraphs $P_1,P_2,\dotsc,P_k$ such that each $P_i$ has prescribed vertices $s_i$ and $t_i$ as endpoints and the union of $P_1,P_2,\dotsc,P_k$ contains all vertices of $G$. In this paper, we introduce bipartite transposition-like graphs, which are inductively constructed from lower ranked bipartite transposition-like graphs. We show that every rank $n$ bipartite transposition-like graph $G$ admit a paired $(n-1)$-to-$(n-1)$ disjoint path cover for all choices of $S=\{s_1,s_2,\dotsc,s_{n-1}\}$ and $T=\{t_1,t_2,\dotsc,t_{n-1}\}$, provided that $S$ is in one partite set of $G$ and $T$ is in the other.

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arXiv2022

Residue sums of Dickson polynomials over finite fields

Given a polynomial with integral coefficients, one can inquire about the possible residues it can take in its image modulo a prime $p$. The sum over the distinct residues can sometimes be computed independent of the prime $p$; for example, Gauss showed that the sum over quadratic residues vanishes modulo a prime. In this paper we provide a closed form for the sum over distinct residues in the image of Dickson polynomials of arbitrary degree over finite fields of odd characteristic, and prove a complete characterization of the size of the image set. Our result provides the first non-trivial classification of such a sum for a family of polynomials of unbounded degree.

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arXiv2022

Probabilistic chip-collecting games with modulo winning conditions

Let $a$, $b$, and $n$ be integers with $0<a<b<n$. In a certain two-player probabilistic chip-collecting game, Alice tosses a coin to determine whether she collects $a$ chips or $b$ chips. If Alice collects $a$ chips, then Bob collects $b$ chips, and vice versa. A player is announced the winner when they have accumulated a number of chips that is a multiple of $n$. In this paper, we settle two conjectures from the literature related to this game.

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arXiv2021

Sum index and difference index of graphs

Let $G$ be a nonempty simple graph with a vertex set $V(G)$ and an edge set $E(G)$. For every injective vertex labeling $f:V(G)\to\mathbb{Z}$, there are two induced edge labelings, namely $f^+:E(G)\to\mathbb{Z}$ defined by $f^+(uv)=f(u)+f(v)$, and $f^-:E(G)\to\mathbb{Z}$ defined by $f^-(uv)=|f(u)-f(v)|$. The sum index and the difference index are the minimum cardinalities of the ranges of $f^+$ and $f^-$, respectively. We provide upper and lower bounds on the sum index and difference index, and determine the sum index and difference index of various families of graphs. We also provide an interesting conjecture relating the sum index and the difference index of graphs.

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arXiv2020

On binomial coefficients associated with Sierpiński and Riesel numbers

In this paper, we investigate the existence of Sierpiński numbers and Riesel numbers as binomial coefficients. We show that for any odd positive integer $r$, there exist infinitely many Sierpiński numbers and Riesel numbers of the form $\binom{k}{r}$. Let $S(x)$ be the number of positive integers $r$ satisfying $1\leq r\leq x$ for which $\binom{k}{r}$ is a Sierpiński number for infinitely many $k$. We further show that the value $S(x)/x$ gets arbitrarily close to 1 as $x$ tends to infinity. Generalizations to base $a$-Sierpiński numbers and base $a$-Riesel numbers are also considered. In particular, we prove that there exist infinitely many positive integers $r$ such that $\binom{k}{r}$ is simultaneously a base $a$-Sierpiński and base $a$-Riesel number for infinitely many $k$.

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arXiv2020

On the properties of fibotomic polynomials

Define the $n$-th fibotomic polynomial to be the product of the monic irredicible factors of the $n$-th Fibonacci polynomial which are not factors of any Fibonacci polynomial of smaller degree. In this paper, we prove a number of properties of the fibotomic polynomials. This includes determining the discriminant of the fibotomic polynomials and the resultant of pairs of fibotomic polynomials. Furthermore, we completely determine the factorization form of the fibotomic polynomials in prime fields. Results are also generalized for the bivariate homogenous fibotomic polynomials.

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arXiv2019

Building matrices with prescribed size and number of invertible submatrices

Given an ordered triple of positive integers $(n,r,b)$, where $1\leq b\leq\binom{n}{r}$, does there exist a matrix of size $r\times n$ with exactly $b$ invertible submatrices of size $r\times r$? Such a matrix is called an $(n,r,b)$-matrix. This question is a stronger version of an open problem in matroid theory raised by Dominic Welsh. In this paper, we prove that an $(n,r,b)$-matrix exists when the corank satisfies $n-r\leq3$, unless $(n,r,b)=(6,3,11)$. Furthermore, we show that an $(n,r,b)$-matrix exists when the rank $r$ is large relative to the corank $n-r$.

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arXiv2018

An Investigation on Partitions with Equal Products

An ordered triple $(s,p,n)$ is called admissible if there exist two different multisets $X=\{x_1,x_2,\dotsc,x_n\}$ and $Y=\{y_1,y_2,\dotsc,y_n\}$ such that $X$ and $Y$ share the same sum $s$, the same product $p$, and the same size $n$. We first count the number of $n$ such that $(s,p,n)$ are admissible for a fixed $s$. We also fully characterize the values $p$ such that $(s,p,n)$ is admissible. Finally, we consider the situation where $r$ different multisets are needed, instead of just two. This project is also related to John Conway's wizard puzzle from the 1960s.

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arXiv2018

On the Domination Number of Permutation Graphs and an Application to Strong Fixed Points

A permutation graph $G_π$ is a simple graph with vertices corresponding to the elements of $π$ and an edge between $i$ and $j$ when $i$ and $j$ are inverted in $π$. A set of vertices $D$ is said to dominate a graph $G$ when every vertex in $G$ is either an element of $D$, or adjacent to an element of $D$. The domination number $γ(G)$ is defined as the cardinality of a minimum dominating set of $G$. A strong fixed point of a permutation $π$ of order $n$ is an element $k$ such that $π^{-1}(j)<π^{-1}(k)$ for all $j π^{-1}(k)$ for all $i>k$. In this article, we count the number of connected permutation graphs on $n$ vertices with domination number $1$ and domination number $\frac{n}{2}$. We further show that for a natural number $k\leq \frac{n}{2}$, there exists a connected permutation graph on $n$ vertices with domination number $k$. We find a closed expression for the number of permutation graphs dominated by a set with two elements, and we find a closed expression for the number of permutation graphs efficiently dominated by any set of vertices. We conclude by providing an application of these results to strong fixed points, proving some conjectures posed on the OEIS.

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arXiv2014

Integrality of relative BPS state counts of toric Del Pezzo surfaces

Relative BPS state counts for log Calabi-Yau surface pairs were introduced by Gross-Pandharipande-Siebert and conjectured by the authors to be integers. For toric Del Pezzo surfaces, we provide an arithmetic proof of this conjecture, by relating these invariants to the local BPS state counts of the surfaces. The latter were shown to be integers by Peng; and more generally for toric Calabi-Yau threefolds by Konishi.

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