arXiv subjects

Enrique Reyes

Publications and source records attributed to Enrique Reyes.

14 recordsLinked to original sources

arXiv2026

Ehrhart Functions of Weighted Lattice Points

This paper studies three different ways to assign weights to the lattice points of a convex polytope and discusses the algebraic and combinatorial properties of the resulting weighted Ehrhart functions and their generating functions and associated rings. These will be called $q$-weighted, $r$-weighted, and $s$-weighted Ehrhart functions, respectively. The key questions we investigate are \emph{When are the weighted Ehrhart series rational functions and which classical Ehrhart theory properties are preserved? And, when are the abstract formal power series the Hilbert series of Ehrhart rings of some polytope?} We prove generalizations about weighted Ehrhart $h^*$-coefficients of $q$-weighted Ehrhart series, and show $q$- and $s$-weighted Ehrhart reciprocity theorems. Then, we show the $q$- and $r$-weighted Ehrhart rings are the (classical) Ehrhart rings of weight lifting polytopes.

math.CO↗

arXiv2025

The v-numbers and linear presentations of ideals of covers of graphs

Let $G$ be a graph and let $J=I_c(G)$ be its ideal of covers. The aims of this work are to study the {\rm v}-number ${\rm v}(J)$ of $J$ and to study when $J$ is linearly presented using combinatorics and commutative algebra. We classify when ${\rm v}(J)$ attains its minimum and maximum possible values in terms of the vertex covers of the graph that satisfy the exchange property. If the cover ideal of a graph has a linear presentation, we express its v-number in terms of the covering number of the graph. If $G$ is unmixed, the graph $\mathcal{G}_J$ of $J$ is the graph whose vertices are the minimal vertex covers of $G$ and whose edges are the pairs $\{C,C'\}$ such that $|C\cup C'|=|C|+1$. We show necessary and sufficient conditions for the graph $\mathcal{G}_J$ of $J$ to be connected. Then, for unmixed König graphs, we classify when $J$ is linearly presented using graph theory, and show some results on Cohen--Macaulay König graphs. If $G$ is unmixed, it is shown that the columns of the linear syzygy matrix of $J$ are linearly independent if and only if $\mathcal{G}_J$ has no strong $3$-cycles. One of our main theorems shows that if $G$ is unmixed and has no induced $4$-cycles, then $J$ is linearly presented. For unmixed graphs without $3$- and $5$-cycles, we classify combinatorially when $J$ is linearly presented.

math.AC↗

arXiv2021

Induced matchings and the v-number of graded ideals

We give a formula for the v-number of a graded ideal that can be used to compute this number. Then we show that for the edge ideal $I(G)$ of a graph $G$ the induced matching number of $G$ is an upper bound for the v-number of $I(G)$ when $G$ is very well-covered, or $G$ has a simplicial partition, or $G$ is well-covered connected and contain neither $4$- nor $5$-cycles. In all these cases the v-number of $I(G)$ is a lower bound for the regularity of the edge ring of $G$. We classify when the upper bound holds when $G$ is a cycle, and classify when all vertices of a graph are shedding vertices to gain insight on $W_2$-graphs.

math.AC↗

arXiv2021

Unmixedness of some weighted oriented graphs

Let $D=(G,\mathcal{O},w)$ be a weighted oriented graph whose edge ideal is $I(D)$. In this paper, we characterize the unmixed property of $I(D)$ for each one of the following cases: $G$ is an $SCQ$ graph; $G$ is a chordal graph; $G$ is a simplicial graph; $G$ is a perfect graph; $G$ has no $4$- or $5$-cycles; $G$ is a graph without $3$- and $5$-cycles; and ${\rm girth}(G)\geqslant 5$.

math.CO↗

arXiv2021

Gorenstein homogeneous subrings of graphs

Let $G=(V,E)$ be a connected simple graph, with $n$ vertices such that $S$ is its homogeneous monomial subring. We prove that if $S$ is normal and Gorenstein, then $G$ is unmixed with cover number $\lceil\frac{n}{2}\rceil$ and $G$ has a strong $\lceil\frac{n}{2}\rceil$-$τ$-reduction. Furthermore, if $n$ is even, then we show that $G$ is bipartite. Finally, if $S$ is normal and $G$ is unmixed whose cover number is $\lceil\frac{n}{2}\rceil$, we give sufficient conditions for $S$ to be Gorenstein.

math.CO↗

arXiv2019

Unmixed and Cohen--Macaulay weighted oriented König graphs

Let $D$ be a weighted oriented graph, whose underlying graph is $G$, and let $I(D)$ be its edge ideal. If $G$ has no $3$-, $5$-, or $7$-cycles, or $G$ is König, we characterize when $I(D)$ is unmixed. If $G$ has no $3$- or $5$-cycles, or $G$ is König, we characterize when $I(D)$ is Cohen--Macaulay. We prove that $I(D)$ is unmixed if and only if $I(D)$ is Cohen--Macaulay when $G$ has girth greater than $7$ or $G$ is König and has no $4$-cycles.

math.AC↗

arXiv2018

Edge ideals of oriented graphs

Let $\mathcal{D}$ be a weighted oriented graph and let $I(\mathcal{D})$ be its edge ideal. Under a natural condition that the underlying (undirected) graph of $\mathcal{D}$ contains a perfect matching consisting of leaves, we provide several equivalent conditions for the Cohen-Macaulayness of $I(\mathcal{D})$. We also completely characterize the Cohen-Macaulayness of $I(\mathcal{D})$ when the underlying graph of $\mathcal{D}$ is a bipartite graph. When $I(\mathcal{D})$ fails to be Cohen-Macaulay, we give an instance where $I(\mathcal{D})$ is shown to be sequentially Cohen-Macaulay.

math.AC↗

arXiv2017

Monomial ideals of weighted oriented graphs

Let I=I(D) be the edge ideal of a weighted oriented graph D. We determine the irredundant irreducible decomposition of I. Also, we characterize the associated primes and the unmixed property of I. Furthermore, we give a combinatorial characterization for the unmixed property of I, when D is bipartite, D is a whisker or D is a cycle. Finally, we study the Cohen-Macaulay property of I.

math.AC↗

arXiv2015

On well-covered, vertex decomposable and Cohen-Macaulay graphs

Let $G=(V,E)$ be a graph. If $G$ is a König graph or $G$ is a graph without 3-cycles and 5-cycle, we prove that the following conditions are equivalent: $Δ_{G}$ is pure shellable, $R/I_Δ$ is Cohen-Macaulay, $G$ is unmixed vertex decomposable graph and $G$ is well-covered with a perfect matching of König type $e_{1},...,e_{g}$ without square with two $e_i$'s. We characterize well-covered graphs without 3-cycles, 5-cycles and 7-cycles. Also, we study when graphs without 3-cycles and 5-cycles are vertex decomposable or shellable. Furthermore, we give some properties and relations between critical, extendables and shedding vertices. Finally, we characterize unicyclic graphs with each one of the following properties: unmixed, vertex decomposable, shellable and Cohen-Macaulay.

math.CO↗

arXiv2014

Graphs and complete intersection toric ideals

Our purpose is to study the family of simple undirected graphs whose toric ideal is a complete intersection from both an algorithmic and a combinatorial point of view. We obtain a polynomial time algorithm that, given a graph $G$, checks whether its toric ideal $P_G$ is a complete intersection or not. Whenever $P_G$ is a complete intersection, the algorithm also returns a minimal set of generators of $P_G$. Moreover, we prove that if $G$ is a connected graph and $P_G$ is a complete intersection, then there exist two induced subgraphs $R$ and $C$ of $G$ such that the vertex set $V(G)$ of $G$ is the disjoint union of $V(R)$ and $V(C)$, where $R$ is a bipartite ring graph and $C$ is either the empty graph, an odd primitive cycle, or consists of two odd primitive cycles properly connected. Finally, if $R$ is $2$-connected and $C$ is connected, we list the families of graphs whose toric ideals are complete intersection.

math.AC↗

arXiv2010

Minimal generators of toric ideals of graphs

Let $I_G$ be the toric ideal of a graph $G$. We characterize in graph theoretical terms the primitive, the minimal, the indispensable and the fundamental binomials of the toric ideal $I_G$.

math.AC↗

arXiv2009

Ring graphs and complete intersection toric ideals

We study the family of graphs whose number of primitive cycles equals its cycle rank. It is shown that this family is precisely the family of ring graphs. Then we study the complete intersection property of toric ideals of bipartite graphs and oriented graphs. An interesting application is that complete intersection toric ideals of bipartite graphs correspond to ring graphs and that these ideals are minimally generated by Groebner bases. We prove that any graph can be oriented such that its toric ideal is a complete intersection with a universal Groebner basis determined by the cycles. It turns out that bipartite ring graphs are exactly the bipartite graphs that have complete intersection toric ideals for any orientation.

math.AC↗

arXiv2008

On 2-partitionable clutters and the MFMC property

We introduce 2-partitionable clutters as the simplest case of the class of $k$-partitionable clutters and study some of their combinatorial properties. In particular, we study properties of the rank of the incidence matrix of these clutters and properties of their minors. A well known conjecture of Conforti and Cornuéjols \cite{ConfortiCornuejols,cornu-book} states: That all the clutters with the packing property have the max-flow min-cut property, i.e. are mengerian. Among the general classes of clutters known to verify the conjecture are: balanced clutters (Fulkerson, Hoffman and Oppenheim \cite{FulkersonHoffmanOppenheim}), binary clutters (Seymour \cite{Seymour}) and dyadic clutters (Cornuéjols, Guenin and Margot \cite{CornuejolsGueninMargot}). We find a new infinite family of 2-partitionable clutters, that verifies the conjecture. On the other hand we are interested in studying the normality of the Rees algebra associated to a clutter and possible relations with the Conforti and Cornuéjols conjecture. In fact this conjecture is equivalent to an algebraic statement about the normality of the Rees algebra \cite{rocky}.

math.AC↗

arXiv2007

Cohen-Macaulay, Shellable and unmixed clutters with a perfect matching of König type

Let $\mathcal{C}$ be a clutter with a perfect matching $e_1,...,e_g$ of König type and let $Δ_\mathcal{C}$ be the Stanley-Reisner complex of the edge ideal of $\mathcal{C}$. If all c-minors of $\mathcal{C}$ have a free vertex and $\mathcal{C}$ is unmixed, we show that $Δ_\mathcal{C}$ is pure shellable. We are able to describe, in combinatorial and algebraic terms, when $Δ_\mathcal{C}$ is pure. If $\mathcal{C}$ has no cycles of length 3 or 4, then it is shown that $Δ_\mathcal{C}$ is pure if and only if $Δ_\mathcal{C}$ is pure shellable (in this case $e_i$ has a free vertex for all $i$), and that $Δ_\mathcal{C}$ is pure if and only if for any two edges $f_1,f_2$ of $\mathcal{C}$ and for any $e_i$, one has that $f_1\cap e_i\subset f_2\cap e_i$ or $f_2\cap e_i\subset f_1\cap e_i$. It is also shown that this ordering condition implies that $Δ_\mathcal{C}$ is pure shellable, without any assumption on the cycles of $\mathcal{C}$. Then we prove that complete admissible uniform clutters and their Alexander duals are unmixed. In addition, the edge ideals of complete admissible uniform clutters are facet ideals of shellable simplicial complexes, they are Cohen-Macaulay, and they have linear resolutions. Furthermore if $ \mathcal{C}$ is admissible and complete, then $\mathcal{C}$ is unmixed. We characterize certain conditions that occur in a Cohen-Macaulay criterion for bipartite graphs of Herzog and Hibi, and extend some results of Faridi--on the structure of unmixed simplicial trees--to clutters with the König property without 3-cycles or 4-cycles.

math.AC↗