arXiv subjects

Mohsen Aliabadi

Publications and source records attributed to Mohsen Aliabadi.

At least 19 recordsLinked to original sources

arXiv2026

A $2/3$ Bound for Vizing's Conjecture

Vizing's conjecture, dating back to 1963, asserts that \[ γ(G\mathbin{\square}H) \geq γ(G)γ(H) \] for all finite graphs $G$ and $H$, where $γ$ denotes the domination number and $\square$ denotes the Cartesian product. In 2000, Clark and Suen proved the universal bound \[ γ(G\mathbin{\square}H) \geq \frac{1}{2}γ(G)γ(H). \] Recently, Steiner obtained the first constant-factor improvement of the Clark--Suen bound, proving that \[ γ(G\mathbin{\square}H) \geq \frac{5+\sqrt{73}}{24}γ(G)γ(H) \approx 0.5643\,γ(G)γ(H). \] In this paper, we further improve the universal constant by proving that \[ γ(G\mathbin{\square}H) \geq \frac{2}{3}γ(G)γ(H) \] for all finite graphs $G$ and $H$. Thus, we raise the best known universal constant in the approximate form of Vizing's conjecture from $(5+\sqrt{73})/24$ to $2/3$.

math.CO↗

arXiv2026

A counterexample to the Chung-Graham-Spiro gap-set conjecture

Chung, Graham, and Spiro introduced slow Fibonacci walks and used them to partition the integers $n\ge2$ into two sequences, the down-integers and the up-integers. They studied the local spacing of these two sequences and conjectured that their $\ell$-step gap sets agree for every $\ell\ge1$. We show that the conjecture fails at $\ell=4$ by proving \[ 9\in U_4\setminus D_4 . \]

math.NT↗

arXiv2026

On partially matchable subspaces in a field extension

We develop a theory of partial matchings between finite-dimensional subspaces $A,B$ in a field extension $K \subsetneq L$, providing structural, quantitative, and extremal results for their deficiency. Our main results include (1) a characterization of those pairs $(A,B)$ that are partially matchable up to a specified defect, (2) a decomposition theorem for pairs $(A,B)$ having positive deficiency, (3) an existence criterion for pairs having a prescribed dimension and satisfying a deficiency bound, and (4) a formula for the maximum attainable deficiency at each dimension, assuming $1 \notin B$ and the extension has a proper nontrivial intermediate field of finite $K$-dimension. We use these results to recover and extend various parts of this area of matching theory. Our approach blends algebraic techniques with tools from matroidal transversal theory, and utilizes a linearized version of the $e$-transform from additive number theory.

math.CO↗

arXiv2026

Matchings and product growth in modular abelian independence groups

We unify two matching theories, one for finite subsets of groups and the other for finite-dimensional subspaces in a field extension. To achieve this, we study groups equipped with a compatible finitary matroid structure, termed here independence groups. Applying Rado's independent transversal theorem, we derive necessary and sufficient rank criteria for matchability between finite-rank sets. In the setting of a modular abelian independence group $G$, we develop an analogue of the $e$-transform from additive number theory, derive structural matching criteria, and characterize a global matching property by the absence of a submonoid $H$ satisfying $1<ρ(H)<ρ(G)$ and $ρ(H)<\infty$, where $ρ$ denotes rank. Examples of modular abelian independence groups are given and examined in the matching context. Arising from this matching theory, but formulated without any reference to it, is a product-growth bound that generalizes the Cauchy--Davenport theorem: we define a parameter $μ(G)$ and prove that $ρ(XY)\geq \min\{μ(G),ρ(X)+ρ(Y)-1\}$ for all nonempty finite-rank subsets $X,Y$ of $G$. Furthermore, $ρ(XY)$ is shown to be controlled from below by a submonoid of $G$ that stabilizes a flat, a phenomenon reminiscent of Kneser's theorem.

math.CO↗

arXiv2026

Extremal Chowla sets and their linear analogues: A human-AI mathematical investigation using Co-Scientist

We introduce an extremal invariant associated with Chowla-type order conditions in finite groups. A nonempty subset $S$ of a finite group $G$ is called a Chowla set if every element of $S$ has order greater than $|S|$, and we write $C(G)$ for the maximum cardinality of such a set. We first show that $C(G)$ is determined by the distribution of element orders in $G$. For cyclic groups, we derive an exact divisor formula and characterize the integers $n$ for which $C(\mathbb{Z}/n\mathbb{Z})=φ(n)$. We prove that $\liminf_{n\to\infty}C(\mathbb{Z}/n\mathbb{Z})/φ(n)=1$, whereas $\limsup_{n\to\infty}C(\mathbb{Z}/n\mathbb{Z})/φ(n)=\infty$, and we determine the corresponding lower and upper limits under normalization by $n$. For finite abelian groups, we obtain an explicit formula in terms of the invariant-factor decomposition, together with a closed formula for finite abelian $p$-groups. We then develop a linear analogue for finite field extensions. A nonzero $K$-subspace $A$ of an extension $L/K$ is called a Chowla subspace if $[K(a):K]>\dim_K A$ for every nonzero $a\in A$. Since this condition depends on $\dim_K A$, it does not generally require every nonzero element of $A$ to generate $L$ over $K$. Nevertheless, when $L/K$ is finite and separable, we prove the exact formula $C(L/K)=[L:K]-d_{\max}(L/K)$, where $d_{\max}(L/K)$ is the largest degree over $K$ of a proper intermediate field. For finite fields, we give a direct proof in every degree using a normal-basis construction. This work was developed through an expert-guided human-AI collaboration. A reasoning-focused configuration of Co-Scientist was used to explore examples and potential proof strategies. The authors formulated the problem, independently verified and completed all arguments, and wrote the final proofs.

math.NT↗

arXiv2026

Higher-power inverse functional identities and Frobenius collision obstructions

Let $D$ be a division ring, let $n\geq 2$, and let $f,g:D\to D$ be additive maps satisfying $f(x)x^{-1}+x^n g(x^{-1})=0$ for all $x\in D^\times$. We establish general vanishing criteria and classify the Frobenius-type obstructions over fields. If $\mathbb{F}_q\subseteq Z(D)$, every additive map $D\to D$ admits a canonical decomposition into $\mathbb{F}_q^\times$-weight components, and the identity pairs precisely the weights $r,s$ satisfying $r+s\equiv n+1\pmod{q-1}$. Consequently, for $q=p^m$, the $\mathbb{F}_q$-vector space $\mathcal{S}_n(\mathbb{F}_q)$ of solutions over $\mathbb{F}_q$ satisfies $\dim_{\mathbb{F}_q}\mathcal{S}_n(\mathbb{F}_q)=\left|\left\{(i,j):0\leq i,j 0$, whenever $(p-1)\nmid(n-1)$. In characteristic two, a generalized-polynomial reduction and an inverse-free identity yield complete vanishing for $n=2$ on every noncommutative division ring. More generally, if $[D:Z(D)]=\infty$, every solution vanishes when the center is infinite. If $Z(D)=\mathbb{F}_q$ is finite, a graded refinement proves the same conclusion for $2\leq n\leq q-1$. The remaining finite-center and centrally finite cases are isolated explicitly.

math.RA↗

arXiv2026

Odd-cycle defects in the Alon-Friedland bound

Let $G$ be a finite simple graph. We study perfect matchings through two complementary viewpoints: reductions to bipartite permanent terms and the directed cycle covers counted by the ordinary adjacency permanent. The main identity is an explicit odd-cycle-indexed form of the classical cycle-cover expansion: it separates $\perfmat(G)^2$, the even-cycle-cover contribution coming from superposing two perfect matchings, from the contribution of cycle covers containing odd cycles. This gives a nonnegative odd-cycle defect $δ_{\mathrm{odd}}(G)$. Combining the identity with the Bregman--Minc inequality yields a structural refinement of the Alon--Friedland degree-sequence bound. We prove product, positivity, and fractional-perfect-matching interpretations for the defect; show that for $K_{2n}$ the defect asymptotically accounts for almost the entire Bregman--Minc target; and derive from this a derangement identity. We also study near equality in the Alon--Friedland bound: we classify all one-edge perturbations of the extremal graphs, record the uniform obstruction coming from $K_4$, and formulate a sharp bounded-degree gap problem whose two natural candidate extremal graphs cross between maximum degrees $9$ and $10$.

math.CO↗

arXiv2026

Decomposition theorems for unmatchable pairs in groups and field extensions

A theory of matchings for finite subsets of an abelian group, introduced in connection with a conjecture of Wakeford on canonical forms for homogeneous polynomials, has since been extended to the setting of field extensions and to that of matroids. Earlier approaches have produced numerous criteria for matchability and unmatchability, but have offered little structural insight. In this paper, we develop parallel structure theorems which characterize unmatchable pairs in both abelian groups and field extensions. Our framework reveals analogous obstructions to matchability: nearly periodic decompositions of sets in the group setting correspond to decompositions of subspaces involving translates of a subfield in the linear setting. This perspective not only recovers previously known results, but also leads to new matching criteria and guarantees the existence of nontrivial unmatchable pairs.

math.CO↗

arXiv2026

A nonabelian twist on differences of bijections

Hall's theorem on differences of bijections characterizes the multisets $$ \{a_1,\ldots,a_{|G|}\} $$ in a finite abelian group $G$ that can be written in the form $$ a_i=b_i-c_i, $$ where both $b_1,\ldots,b_{|G|}$ and $c_1,\ldots,c_{|G|}$ are enumerations of $G$. The necessary and sufficient condition is the zero-sum condition $$ a_1+\cdots+a_{|G|}=0. $$ This paper studies the corresponding problem for finite nonabelian groups, with differences replaced by quotients. Thus we ask when a multiset $A$ of cardinality $|G|$ can be represented as $$ A=\{b(i)c(i)^{-1}:1\le i\le |G|\}, $$ where $b$ and $c$ are bijections onto $G$. Passing to the abelianization gives a necessary condition, namely that the product of the images of the elements of $A$ is trivial in $ G_{\rm ab}. $ We show that this condition is not sufficient in general, even when the elements of $A$ admit an ordering whose product is the identity in $G$. The main structural result is a cycle-tiling criterion: quotient-realizability is equivalent to a decomposition of $A$ into product-one words whose partial-product sets tile $G$ by right translates. The use of permutation cycles is standard, but the criterion translates quotient-realizability into an exact tiling condition. We then use this criterion to construct a counterexample in $ S_3, $ and we extend the same obstruction to infinitely many finite nonabelian groups.

math.GR↗

arXiv2026

A counterexample to a basis conjecture of Brualdi, Friedland, and Pothen

Brualdi, Friedland, and Pothen studied sparse bases of row spaces and proposed a rank-intersection criterion for elementary row-space vectors of sparse generic matrices. We give a $4\times 8$ sparse generic counterexample. The example has four elementary vectors in its row space whose zero sets satisfy all the proposed rank-intersection inequalities, but the four vectors satisfy a nontrivial linear relation and hence do not form a basis. The construction also explains the obstruction: the proposed inequalities see only ranks of column sets indexed by intersections of zero sets, while linear independence of the corresponding elementary vectors is governed by the relative position of the hyperplanes annihilated by their coefficient vectors.

math.CO↗

arXiv2026

Characterization of matchable sets and subspaces via Dyson transforms

A \emph{matching} from a finite subset $ A $ of an abelian group $ G $ to another subset $ B $ is a bijection $ f : A \to B $ such that $ af(a) \notin A $ for all $ a \in A $. The study of matchings began in the 1990s and was motivated by a conjecture of E.~K.~Wakeford on canonical forms for homogeneous polynomials. The theory was later extended to the linear setting of vector subspaces over field extensions, and then to matroids. In this paper, we investigate the existence and structure of matchings in both abelian groups and field extensions. Using Dyson's $ e $-transform, a tool from additive combinatorics, along with a linear analogue which is introduced in this paper, we establish characterization theorems for matchable sets and subspaces. Several applications are given to demonstrate the effectiveness of these theorems as standalone tools. Throughout, we highlight the parallels between the group-theoretic and linear perspectives.

math.CO↗

arXiv2026

Diagonal parity and loop toggling for symmetric matrices over $\mathbb F_2$

Let $M$ be a symmetric matrix over $\mathbb F_2$, and let $\diag(M)$ be its diagonal vector. It is known that \[ \diag(M)\in \Img(M). \] Thus the affine system $Mx=\diag(M)$ is always solvable. We strengthen this existence statement to a parity rigidity theorem: every solution satisfies \[ \diag(M)^T x\equiv \rank(M)\pmod 2 . \] For graph matrices this gives a common extension of Sutner's odd-domination theorem and Batal's parity theorem from closed-neighborhood matrices $A(G)+I$ to arbitrary partially looped graph matrices $A(G)+D$. We also study how rank and nullity change when loops are toggled. Algebraically, simultaneous loop toggling on the support of a vector $u$ is the diagonal rank-one update $M\mapsto M+uu^T$. We prove an exact three-case rank and nullity formula for this update. Finally, for rooted trees with arbitrary binary diagonal labels, we give a finite-state boundary recursion using affine subspaces of $\mathbb F_2^2$. This recursion counts all generalized odd-domination patterns and implies eventual quasigeometric nullity formulas for complete rooted trees with eventually periodic depth labels.

math.CO↗

arXiv2026

Paving matroids that are not sparse paving

The Mayhew--Newman--Welsh--Whittle conjecture predicts that asymptotically almost all matroids are sparse paving. We study the gap between paving and sparse paving matroids at the logarithmic scale. Let $p_n$ be the number of paving matroids on $[n]$, let $sp_n$ be the number of sparse paving matroids on $[n]$, and let $sp_{n,r}$ be the number of rank-$r$ sparse paving matroids on $[n]$. We prove that \[ p_n-sp_n\ge sp_{n,\lfloor n/2\rfloor}^{1-o(1)}. \] Thus the paving matroids that are not sparse paving are themselves logarithmically large. The construction prescribes one hyperplane larger than the rank and then counts stable sets in an induced subgraph of a Johnson graph. We also give amplified versions obtained by varying the large hyperplane and by prescribing distance-six families of large hyperplanes.

math.CO↗

arXiv2026

Matchings in Matroids over Abelian Groups, III

This paper develops matroidal analogues of classical results on matchings in abelian groups. By embedding matroid ground sets in an abelian group, we introduce base matchings between matroid bases, recover the group-theoretic setting in the uniform matroid case, and derive structural and combinatorial criteria for their existence. Our main focus is on paving matroids. We prove self-matchability for paving matroids, extend asymmetric matchability results using the hyperplane-nullity parameter, and show that stressed hyperplanes provide a natural route to matchability through relaxation.

math.CO↗

arXiv2026

Structure and Decomposition of Deltoids in Abelian Groups

Deltoids provide a natural framework for studying defective (partial) matchings in abelian groups, and we develop both structure and existence results in this setting. Given finite subsets $A$ and $B$ of an abelian group $G$, a matching is a bijection $f:A\to B$ such that $af(a)\notin A$ for all $a\in A$, a definition motivated by the study of canonical forms for symmetric tensors. We provide necessary and sufficient conditions for the existence of a partial matching with any prescribed defect, and then describe the minimal unavoidable defect for a pair $(A,B)$. We also define and examine a defective version of Chowla sets in the matching context. We prove a structure theorem identifying obstructions to the existence of partial matchings with small defect. Finally, within the deltoid setup, we establish max-min results on the partitioning of $A$ and $B$ into left- and right-admissible sets. Our tools mix results from transversal theory with ideas from additive number theory.

math.CO↗

arXiv2025

The weak acyclic matching property in abelian groups

A matching from a finite subset $A\subset\mathbb{Z}^n$ to another subset $B\subset\mathbb{Z}^n$ is a bijection $f : A \rightarrow B$ with the property that $a+f(a)$ never lies in $A$. A matching is called acyclic if it is uniquely determined by its multiplicity function. Alon et al. established the acyclic matching property for $\mathbb{Z}^n$, which was later extended to all abelian torsion-free groups. In a prior work, the authors of this paper settled the acyclic matching property for all abelian groups. The objective of this note is to explore a related concept, known as the weak acyclic matching property, within the context of abelian groups.

math.CO↗

arXiv2025

Matchings in matroids over abelian groups, II

The concept of matchings originated in group theory to address a linear algebra problem related to canonical forms for symmetric tensors. In an abelian group $(G,+)$, a matching is a bijection $f: A \to B$ between two finite subsets $A$ and $B$ of $G$ such that $a + f(a) \notin A$ for all $a \in A$. A group $G$ has the matching property if, for every two finite subsets $A, B \subset G$ of the same size with $0 \notin B$, there exists a matching from $A$ to $B$. In prior work [5], matroid analogues of results concerning matchings in groups were introduced and established. This paper serves as a sequel, extending that line of inquiry by investigating sparse paving, panhandle, and Schubert matroids through the lens of matchability. While some proofs draw upon earlier findings on the matchability of sparse paving matroids, the paper is designed to be self-contained and accessible without reference to the preceding sequel. Our approach combines tools from both matroid theory and additive number theory.

math.CO↗

arXiv2024

Classifying abelian groups through acyclic matchings

The inquiry into identifying sets of monomials that can be eliminated from a generic homogeneous polynomial via a linear change of coordinates was initiated by E. K. Wakeford. This linear algebra problem prompted C. K. Fan and J. Losonczy to introduce the notion of acyclic matchings in the additive group $\mathbb{Z}^n$, subsequently extended to abelian groups by the latter author. Alon, Fan, Kleitman, and Losonczy established the acyclic matching property for $\mathbb{Z}^n$. This note aims to classify all abelian groups with respect to the acyclic matching property.

math.CO↗