arXiv subjects

Ignacio García-Marco

Publications and source records attributed to Ignacio García-Marco.

At least 19 recordsLinked to original sources

arXiv2026

Rings are $χ$-bounded

We prove that there is a function $f:\mathbb N\to\mathbb N$ such that $χ(Γ(R))\leq f(ω(Γ(R)))$ for the zero-divisor graph of any ring $R$, if the clique number is finite. On the one hand, this consolidates a disproved conjecture of Beck from 1988, claiming $χ(Γ(R))=ω(Γ(R))$ for unital commutative rings. While previous counterexamples satisfy $χ(Γ(R))\leq ω(Γ(R))+2$, we obtain the lower bound $f\geq k^{Ω(\log k)}$ among finite commutative rings. Finally, we show that neither zero-divisor graphs of finite commutative semirings nor those of finite commutative nonassociative rings are $χ$-bounded.

math.CO↗

arXiv2026

Castelnuovo-Mumford regularity of toric varieties with at most one singular point

We establish upper bounds for the Castelnuovo--Mumford regularity of the coordinate ring of a simplicial projective toric variety with at most one singular point. In the smooth case, our results recover the bound of Herzog and Hibi [Proc. Amer. Math. Soc. 131 (2003), 2641--2647], and therefore the Eisenbud--Goto bound. Furthermore, when the variety has exactly one singular point and dimension at least $3$, we prove that its regularity also satisfies the Eisenbud--Goto bound. The proof combines combinatorial and homological methods: we study the asymptotic behavior of the sumsets associated to the toric variety and relate it to Castelnuovo--Mumford regularity via a Hochster-like formula.

math.AC↗

arXiv2026

Coloring semiminimal Cayley Graphs

In 1978 Babai raised the question whether all semiminimal Cayley graphs have bounded chromatic number. In this paper we show that semiminimal Cayley graphs of abelian and generalized dihedral groups have circular chromatic number at most 4, thus extending a result of Barajas and Serra.

math.CO↗

arXiv2026

Generalized Andrásfai graphs and special Betti diagrams of edge ideals

Edge ideals of graphs were introduced by Villarreal in 1990, and have been the subject of many studies since then. In the same year, Fröberg characterized edge ideals with regularity 2 in combinatorial terms. This result was generalized by Fernández-Ramos and Gimenez to regularity 3 for bipartite graphs. A key ingredient in these results is the particular shape of the Betti diagrams of the edge ideals of the graphs obtained after removing a Hamiltonian cycle from either a complete graph $ K_k$ or a complete bipartite graph $K_{k,k}$. In this work, we consider the family of Generalized Andrásfai graphs ${\rm GA}(t,k)$ with $t\geq 1 $ and $k \geq 2$. This family extends the families of complete graphs, since $K_{k+1} = {\rm GA}(1,k)$, and complete bipartite $k$-regular graphs, since $K_{k,k} = {\rm GA}(2,k)$. We show that the results known for $ K_k$ and $ K_{k,k}$ can be naturally extended to this family. More precisely, when removing a suitable Hamiltonian cycle from ${\rm GA}(t,k)$, the resulting edge ideal has regularity $t+2$, projective dimension $t(k-2)$ and a Betti diagram exhibiting a generalized version of the same special shape.

math.AC↗

arXiv2026

Sensitivity and Hamming graphs

For any $m\geq 3$ we show that the Hamming graph $H(n,m)$ admits an imbalanced partition into $m$ sets, each inducing a subgraph of low maximum degree. This improves previous results by Tandya and by Potechin and Tsang, and disproves the Strong $m$-ary Sensitivity Conjecture of Asensio, García-Marco, and Knauer. On the other hand, we prove their weaker $m$-ary Sensitivity Conjecture by showing that the sensitivity of any $m$-ary function is bounded from below by a polynomial expression in its degree.

math.CO↗

arXiv2025

On the minimum number of Toeplitz factors of a matrix

We disprove a conjecture by Ye and Lim, by showing that there are $3 \times 3$ complex matrices which can't be expressed as the product of two Toeplitz matrices of the same size. We also improve previous estimates by Ye and Lim on the minimum number of Toeplitz matrices needed to factor any $n \times n$ matrix, for low values of $n$.

math.AC↗

arXiv2025

Computational Aspects of the Short Resolution

Let $R:= \Bbbk[x_1,\ldots,x_{n}]$ be a polynomial ring over a field $\Bbbk$, $I \subset R$ be a homogeneous ideal with respect to a weight vector $ω= (ω_1,\ldots,ω_n) \in (\mathbb{Z}^+)^n$, and denote by $d$ the Krull dimension of $R/I$. In this paper we study graded free resolutions of $R/I$ as $A$-module whenever $A :=\Bbbk[x_{n-d+1},\ldots,x_n]$ is a Noether normalization of $R/I$. We exhibit a Schreyer-like method to compute a (non-necessarily minimal) graded free resolution of $R/I$ as $A$-module. When $R/I$ is a $3$-dimensional simplicial toric ring, we describe how to prune the previous resolution to obtain a minimal one. We finally provide an example of a $6$-dimensional simplicial toric ring whose Betti numbers, both as $R$-module and as $A$-module, depend on the characteristic of $\Bbbk$.

math.AC↗

arXiv2025

Toric ideals of graphs minimally generated by a Gröbner basis

Describing families of ideals that are minimally generated by at least one, or by all, of their reduced Gröbner bases is a central topic in commutative algebra. In this paper, we address this problem in the context of toric ideals of graphs. We say that a graph $G$ is an MG-graph if its toric ideal $I_G$ is minimally generated by some Gröbner basis, and a UMG-graph if every reduced Gröbner basis of $I_G$ forms a minimal generating set. We prove that a graph $G$ is a UMG-graph if and only if its toric ideal $I_G$ is a generalized robust ideal (that is, its universal Gröbner basis coincides with its universal Markov basis). Although the class of MG-graphs is not closed under taking subgraphs, we prove that it is hereditary, that is, closed under taking induced subgraphs. In addition, we describe two families of bipartite MG-graphs: ring graphs (which correspond to complete intersection toric ideals, as shown by Gitler, Reyes, and Villarreal) and graphs in which all chordless cycles have the same length. The latter extends a result of Ohsugi and Hibi, which corresponds to graphs whose chordless cycles are all of length $4$.

math.AC↗

arXiv2025

On decoding hyperbolic codes

This work studies several decoding algorithms for hyperbolic codes. We use some previous ideas to describe how to decode a hyperbolic code using the largest Reed-Muller code contained in it or using the smallest Reed-Muller code that contains it. A combination of these two algorithms is proposed when hyperbolic codes are defined by polynomials in two variables. Then, we compare hyperbolic codes and Cube codes (tensor product of Reed-Solomon codes) and propose decoding algorithms of hyperbolic codes based on their closest Cube codes. Finally, we adapt to hyperbolic codes the Geil and Matsumoto's generalization of Sudan's list decoding algorithm.

cs.IT↗

arXiv2025

Proyective Cohen-Macaulay monomial curves and their affine charts

In this paper, we explore when the Betti numbers of the coordinate rings of a projective monomial curve and one of its affine charts are identical. Given an infinite field $k$ and a sequence of relatively prime integers $a_0 = 0 < a_1 < \cdots < a_n = d$, we consider the projective monomial curve $\mathcal{C}\subset\mathbb{P}_k^{\,n}$ of degree $d$ parametrically defined by $x_i = u^{a_i}v^{d-a_i}$ for all $i \in \{0,\ldots,n\}$ and its coordinate ring $k[\mathcal{C}]$. The curve $\mathcal{C}_1 \subset \mathbb A_k^n$ with parametric equations $x_i = t^{a_i}$ for $i \in \{1,\ldots,n\}$ is an affine chart of $\mathcal{C}$ and we denote by $k[\mathcal{C}_1]$ its coordinate ring. The main contribution of this paper is the introduction of a novel (Gröbner-free) combinatorial criterion that provides a sufficient condition for the equality of the Betti numbers of $k[\mathcal{C}]$ and $k[\mathcal{C}_1]$. Leveraging this criterion, we identify infinite families of projective curves satisfying this property. Also, we use our results to study the so-called shifted family of monomial curves, i.e., the family of curves associated to the sequences $j+a_1 < \cdots < j+a_n$ for different values of $j \in \mathbb N$. In this context, Vu proved that for large enough values of $j$, one has an equality between the Betti numbers of the corresponding affine and projective curves. Using our results, we improve Vu's upper bound for the least value of $j$ such that this occurs.

math.AC↗

arXiv2024

Coloring minimal Cayley graphs

In 1978 Babai raised the question whether all minimal Cayley graphs have bounded chromatic number; in 1994 he conjectured a negative answer. In this paper we show that any minimal Cayley graph of a (finitely generated) generalized dihedral or nilpotent group has chromatic number at most 3, while 4 colors are sometimes necessary for soluble groups. On the other hand we address a related question proposed by Babai in 1978 by constructing graphs of unbounded chromatic number that admit a proper edge coloring such that each cycle has some color at least twice. The latter can be viewed as a step towards confirming Babai's 1994 conjecture -- a problem that remains open.

math.CO↗

arXiv2024

Sensitivity of $m$-ary functions and low degree partitions of Hamming graphs

The study of complexity measures of Boolean functions led Nisan and Szegedy to state the sensitivity conjecture in 1994, claiming a polynomial relation between degree and sensitivity. This problem remained unsolved until 2019, when Huang proved the conjecture via an equivalent graph theoretical reformulation due to Gotsman and Linial. We study $m$-ary functions, i.e., functions $f: T^n \rightarrow T$ where $T\subseteq \mathbb{C}$ is a finite alphabet of cardinality $|T| = m $ and extend the notions of degree $\mathrm{deg}(f)$ and sensitivity $s(f)$ to $m$-ary functions and show $s(f)\in O(\mathrm{deg}(f)^2)$. This generalizes results of Nisan and Szegedy. Conversely, we introduce the $m$-ary sensitivity conjecture, claiming a polynomial upper bound for $\mathrm{deg}(f)$ in terms of $s(f)$. Analogously to results of Gotsman and Linial, we provide a formulation of the conjecture in terms of imbalanced partitions of Hamming graphs into low degree subgraphs. Combining this with ideas of Chung, Füredi, Graham and Seymour, we show that for any prime $p$ the bound in the $p$-ary sensitivity conjecture has to be at least quadratic: there exist $p$-ary functions $f$ of arbitrarily large degree and $\mathrm{deg}(f)\in Ω(s(f)^2)$.

math.CO↗

arXiv2022

On the generalized Hamming weights of hyperbolic codes

A hyperbolic code is an evaluation code that improves a Reed-Muller because the dimension increases while the minimum distance is not penalized. We give the necessary and sufficient conditions, based on the basic parameters of the Reed-Muller, to determine whether a Reed-Muller coincides with a hyperbolic code. Given a hyperbolic code, we find the largest Reed-Muller containing the hyperbolic code and the smallest Reed-Muller in the hyperbolic code. We then prove that similarly to Reed-Muller and Cartesian codes, the $r$-th generalized Hamming weight and the $r$-th footprint of the hyperbolic code coincide. Unlike Reed-Muller and Cartesian, determining the $r$-th footprint of a hyperbolic code is still an open problem. We give upper and lower bounds for the $r$-th footprint of a hyperbolic code that, sometimes, are sharp.

cs.IT↗

arXiv2022

On robustness and related properties on toric ideals

A toric ideal is called robust if its universal Gröbner basis is a minimal set of generators, and is called generalized robust if its universal Gröbner basis equals its universal Markov basis (the union of all its minimal sets of binomial generators). Robust and generalized robust toric ideals are both interesting from both a Commutative Algebra and an Algebraic Statistics perspective. However, only a few nontrivial examples of such ideals are known. In this work we study these properties for toric ideals of both graphs and numerical semigroups. For toric ideals of graphs, we characterize combinatorially the graphs giving rise to robust and to generalized robust toric ideals generated by quadratic binomials. As a byproduct, we obtain families of Koszul rings. For toric ideals of numerical semigroups, we determine that one of its initial ideals is a complete intersection if and only if the semigroup belongs to the so-called family of free numerical semigroups. Hence, we characterize all complete intersection numerical semigroups which are minimally generated by one of its Gröbner basis and, as a consequence, all the Betti numbers of the toric ideal and its corresponding initial ideal coincide. Moreover, also for numerical semigroups, we prove that the ideal is generalized robust if and only if the semigroup has a unique Betti element and that there are only trivial examples of robust ideals. We finish the paper with some open questions.

math.AC↗

arXiv2022

Free Resolutions and Generalized Hamming Weights of binary linear codes

In this work, we explore the relationship between free resolution of some monomial ideals and Generalized Hamming Weights (GHWs) of binary codes. More precisely, we look for a structure smaller than the set of codewords of minimal support that provides us some information about the GHWs. We prove that the first and second generalized Hamming weight of a binary linear code can be computed (by means of a graded free resolution) from a set of monomials associated to a binomial ideal related with the code. Moreover, the remaining weights are bounded by the Betti numbers for that set.

cs.IT↗

arXiv2022

Beyond symmetry in generalized Petersen graphs

A graph is a core or unretractive if all its endomorphisms are automorphisms. Well-known examples of cores include the Petersen graph and the graph of the dodecahedron -- both generalized Petersen graphs. We characterize the generalized Petersen graphs that are cores. A simple characterization of endomorphism-transitive generalized Petersen graphs follows. This extends the characterization of vertex-transitive generalized Petersen graphs due to Frucht, Graver, and Watkins and solves a problem of Fan and Xie. Moreover, we study generalized Petersen graphs that are (underlying graphs of) Cayley graphs of monoids. We show that this is the case for the Petersen graph, answering a recent mathoverflow question, for the Desargues graphs, and for the dodecahedron -- answering a question of Knauer and Knauer. Moreover, we characterize the infinite family of generalized Petersen graphs that are Cayley graph of a monoid with generating connection set of size two. This extends Nedela and Škoviera's characterization of generalized Petersen graphs that are group Cayley graphs and complements results of Hao, Gao, and Luo.

math.CO↗

arXiv2021

On sensitivity in bipartite Cayley graphs

Huang proved that every set of more than half the vertices of the $d$-dimensional hypercube $Q_d$ induces a subgraph of maximum degree at least $\sqrt{d}$, which is tight by a result of Chung, Füredi, Graham, and Seymour. Huang asked whether similar results can be obtained for other highly symmetric graphs. First, we present three infinite families of Cayley graphs of unbounded degree that contain induced subgraphs of maximum degree $1$ on more than half the vertices. In particular, this refutes a conjecture of Potechin and Tsang, for which first counterexamples were shown recently by Lehner and Verret. The first family consists of dihedrants and contains a sporadic counterexample encountered earlier by Lehner and Verret. The second family are star graphs, these are edge-transitive Cayley graphs of the symmetric group. All members of the third family are $d$-regular containing an induced matching on a $\frac{d}{2d-1}$-fraction of the vertices. This is largest possible and answers a question of Lehner and Verret. Second, we consider Huang's lower bound for graphs with subcubes and show that the corresponding lower bound is tight for products of Coxeter groups of type $\mathbf{A_n}$, $\mathbf{I_2}(2k+1)$, and most exceptional cases. We believe that Coxeter groups are a suitable generalization of the hypercube with respect to Huang's question. Finally, we show that induced subgraphs on more than half the vertices of Levi graphs of projective planes and of the Ramanujan graphs of Lubotzky, Phillips, and Sarnak have unbounded degree. This gives classes of Cayley graphs with properties similar to the ones provided by Huang's results. However, in contrast to Coxeter groups these graphs have no subcubes.

math.CO↗

arXiv2021

Factorizations of the same length in abelian monoids

Let $\mathcal S \subseteq \mathbb Z^m \oplus T$ be a finitely generated and reduced monoid. In this paper we develop a general strategy to study the set of elements in $\mathcal S$ having at least two factorizations of the same length, namely the ideal $\mathcal L_{\mathcal S}$. To this end, we work with a certain (lattice) ideal associated to the monoid $\mathcal S$. Our study can be seen as a new approach generalizing \cite{chapman:2011}, which only studies the case of numerical semigroups. When $\mathcal S$ is a numerical semigroup we give three main results: (1) we compute explicitly a set of generators of the ideal $\mathcal L_{\mathcal S}$ when $\mathcal S$ is minimally generated by an almost arithmetic sequence; (2) we provide an infinite family of numerical semigroups such that $\mathcal L_{\mathcal S}$ is a principal ideal; (3) we classify the computational problem of determining the largest integer not in $\mathcal L_{\mathcal S}$ as an $\mathcal{NP}$-hard problem.

math.AC↗