arXiv subjects

Lluís Vena

Publications and source records attributed to Lluís Vena.

18 recordsLinked to original sources

arXiv2026

Alon-Tarsi for hypergraphs

Given a hypergraph $H=(V,E)$, define for every edge $e\in E$ a linear expression with arguments corresponding to the vertices. Next, let the polynomial $p_H$ be the product of such linear expressions for all edges. Our main goal is to find a relationship between the Alon-Tarsi number of $p_H$ and the edge density of $H$. We prove that $AT(p_H)=\lceil \mathrm{ed}(H)\rceil+1$ if all the coefficients in $p_H$ are equal to $1$ and the base field has characteristic zero. Our main result is that, over an arbitrary field, if on every edge the coefficients are not all equal, then they can be permuted within the edges so that for the resulting polynomial $p_H^\prime$, $AT(p_H^\prime)\leq 2\lceil \mathrm{ed}(H)\rceil+1$ holds. We conjecture that this bound holds for every hypergraph polynomial without permuting its coefficients. If this were true, then in particular a significant generalization of the famous 1-2-3 Conjecture would follow.

math.CO↗

arXiv2026

Graham conjecture on small sets in abelian groups

A famous conjecture of Graham asserts that every set $A \subseteq \mathbb{Z}_p \setminus \{0\}$ can be ordered so that all partial sums are distinct. Although this conjecture was recently proved for sufficiently large primes by Pham and Sauermann in~\cite{PM} (combined with earlier results of \cite{BBKMM}), it remains open for general abelian groups, even in the cyclic case $\mathbb{Z}_k$. In this paper, using a recursive approach, we investigate the sequenceability of subsets $A$ in generic abelian groups for small values of $|A|$. We prove that any subset $A \subseteq G\setminus\{0\}$ with $|A| \leq 20$ is sequenceable where previously it was known only for $|A|\leq 9$. This bound is improved to $|A| \leq 22$ for zero-sum subsets. Finally, regarding the related CMPP conjecture, we show that zero-sum subsets without inverse pairs are sequenceable for $|A| \leq 23$.

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arXiv2025

Reconstructing graphs and their connectivity using graphlets

Graphlets are subgraphs rooted at a fixed vertex. The number of occurrences of graphlets aligned to a particular vertex, called graphlet degree sequence (gds), gives a topological description of the surrounding of the analyzed vertex. Graphlet degree distribution (gdd) of a graph is a matrix containing graphlet degree sequence for all vertices in the given graph. A long standing open problem called reconstruction conjecture (RC) asks whether the structure of a graph is uniquely determined by the multiset of its vertex-deleted subgraphs. Graphlet degree distribution up to size (n - 1), (<= n - 1)-gdd, gives more information to reconstruct the graph and we use it to reconstruct any graph having a unique almost-asymmetric vertex-deleted subgraph, where almost-asymmetric means that at most one automorphism orbit has size larger than one. Moreover, we prove that any graph containing a vertex-cut of size 1 or any graph of order n having a vertex with degree at most 2 or at least n-2 is reconstructible from its (<= n - 1)-gdd, which expands results shown in the standard RC. We also discuss the relation between gdd and graph connectivity and the conditions on (<= 3)-gdd, whose breaking means that no graph with such gdd exists.

math.CO↗

arXiv2023

On the extremal families for the Kruskal--Katona theorem

In \cite[Serra, Vena, Extremal families for the Kruskal-Katona theorem]{sv21}, the authors have shown a characterization of the extremal families for the Kruskal-Katona Theorem. We further develop some of the arguments given in \cite{sv21} and give additional properties of these extremal families. Füredi-Griggs/Mörs theorem from 1986/85 \cite{furgri86,mors85} claims that, for some cardinalities, the initial segment of the colexicographical is the unique extremal family; we extend their result as follows: the number of (non-isomorphic) extremal families strictly grows with the gap between the last two coefficients of the $k$-binomial decomposition. We also show that every family is an induced subfamily of an extremal family, and that, somewhat going in the opposite direction, every extremal family is close to being the inital segment of the colex order; namely, if the family is extremal, then after performing $t$ lower shadows, with $t=O(\log(\log n))$, we obtain the initial segment of the colexicographical order. We also give a ``fast'' algorithm to determine whether, for a given $t$ and $m$, there exists an extremal family of size $m$ for which its $t$-th lower shadow is not yet the initial segment in the colexicographical order. As a byproduct of these arguments, we give yet another characterization of the families of $k$-sets satisfying equality in the Kruskal--Katona theorem. Such characterization is, at first glance, less appealing than the one in \cite{sv21}, since the additional information that it provides is indirect. However, the arguments used to prove such characterization provide additional insight on the structure of the extremal families themselves.

math.CO↗

arXiv2023

Homomorphisms between graphs embedded on surfaces

We extend the notion of graph homomorphism to cellularly embedded graphs (maps) by designing operations on vertices and edges that respect the surface topology; we thus obtain the first definition of map homomorphism that preserves both the combinatorial structure (as a graph homomorphism) and the topological structure of the surface (in particular, orientability and genus). Notions such as the core of a graph and the homomorphism order on cores are then extended to maps. We also develop a purely combinatorial framework for various topological features of a map such as the contractibility of closed walks, which in particular allows us to characterize map cores. We then show that the poset of map cores ordered by the existence of a homomorphism is connected and, in contrast to graph homomorphisms, does not contain any dense interval (so it is not universal for countable posets). Finally, we give examples of a pair of cores with an infinite number of cores between them, an infinite chain of gaps, and arbitrarily large antichains with a common homomorphic image.

math.CO↗

arXiv2023

Extremal families for the Kruskal--Katona theorem

Given a family $S$ of $k$--subsets of $[n]$, its lower shadow $Δ(S)$ is the family of $(k-1)$--subsets which are contained in at least one set in $S$. The celebrated Kruskal--Katona theorem gives the minimum cardinality of $Δ(S)$ in terms of the cardinality of $S$. Füredi and Griggs (and Mörs) showed that the extremal families for this shadow minimization problem in the Boolean lattice are unique for some cardinalities and asked for a general characterization of these extremal families. In this paper we prove a new combinatorial inequality from which yet another simple proof of the Kruskal--Katona theorem can be derived. The inequality can be used to obtain a characterization of the extremal families for this minimization problem, giving an answer to the question of Füredi and Griggs. Some known and new additional properties of extremal families can also be easily derived from the inequality.

math.CO↗

arXiv2022

Irreducibility of the Tutte polynomial of an embedded graph

We prove that the ribbon graph polynomial of a graph embedded in an orientable surface is irreducible if and only if the embedded graph is neither the disjoint union nor the join of embedded graphs. This result is analogous to the fact that the Tutte polynomial of a graph is irreducible if and only if the graph is connected and non-separable.

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arXiv2021

Big Ramsey degrees and forbidden cycles

Using the Carlson-Simpson theorem, we give a new general condition for a structure in a finite binary relational language to have finite big Ramsey degrees

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arXiv2018

On the circular chromatic number of a subgraph of the Kneser graph

Let $n,k,r$ be positive integers with $n \geq rk$ and $r \geq 2$. Consider a circle $C$ with~$n$ points~$1,\ldots,n$ in clockwise order. The $r$-stable \emph{interlacing graph} $\text{IG}_{n,k}^{(r)}$ is the graph with vertices corresponding to $k$-subsets $S$ of $\{1,...,n\}$ such that any two distinct points in~$S$ have distance at least~$r$ around the circle, and edges between~$k$-subsets $P$ and $Q$ if they \emph{interlace}: after removing the points in~$P$ from $C$, the points in~$Q$ are in different connected components. In this paper we prove that the circular chromatic number of $\text{IG}_{n,k}^{(r)}$ is equal to $ n/k $ (hence the chromatic number is $\lceil n/k \rceil$) and that its circular clique number is also $ n/k $. Furthermore, we show that its independence number is $\binom{n-(r-1)k-1}{k-1}$, thereby strengthening a result by Talbot.

math.CO↗

arXiv2018

A Tutte polynomial for maps II: the non-orientable case

We construct a new polynomial invariant of maps (graphs embedded in a compact surface, orientable or non-orientable), which contains as specializations the Krushkal polynomial, the Bollobás--Riordan polynomial, the Las Vergnas polynomial, and their extensions to non-orientable surfaces, and hence in particular the Tutte polynomial. Other evaluations include the number of local flows and local tensions taking non-identity values in a given finite group.

math.CO↗

arXiv2017

On a problem by Shapozenko on Johnson graphs

The Johnson graph $J(n,m)$ has the $m$--subsets of $\{1,2,\ldots,n\}$ as vertices and two subsets are adjacent in the graph if they share $m-1$ elements. Shapozenko asked about the isoperimetric function $μ_{n,m}(k)$ of Johnson graphs, that is, the cardinality of the smallest boundary of sets with $k$ vertices in $J(n,m)$ for each $1\le k\le {n\choose m}$. We give an upper bound for $μ_{n,m}(k)$ and show that, for each given $k$ such that the solution to the Shadow Minimization Problem in the Boolean lattice is unique, and each sufficiently large $n$, the given upper bound is tight. We also show that the bound is tight for the small values of $k\le m+1$ and for all values of $k$ when $m=2$.

math.CO↗

arXiv2016

Counting configuration-free sets in groups

We provide new examples of the asymptotic counting for the number of subsets on groups of given size which are free of certain configurations. These examples include sets without solutions to equations in non-abelian groups, and linear configurations in abelian groups defined from group homomorphisms. The results are obtained by combining the methodology of hypergraph containers joint with arithmetic removal lemmas. As a consequence, random counterparts are presented as well.

math.CO↗

arXiv2015

On linear configurations in subsets of compact abelian groups, and invariant measurable hypergraphs

We prove an arithmetic removal result for all compact abelian groups, generalizing a finitary removal result of Král', Serra and the third author. To this end, we consider infinite measurable hypergraphs that are invariant under certain group actions, and for these hypergraphs we prove a symmetry-preserving removal lemma, which extends a finitary result of the same name by the second author. We deduce our arithmetic removal result by applying this lemma to a specific type of invariant measurable hypergraph. As a direct application, we obtain the following generalization of Szemerédi's theorem: for any compact abelian group $G$, any measurable set $A\subset G$ with Haar probability $μ(A)\geqα>0$ satisfies $$\int_G\int_G\; 1_A\big(x\big)\; 1_A\big(x+r\big) \cdots 1_A\big(x+(k-1)r\big) \; dμ(x) dμ(r) \geq c,$$ where the constant $c=c(α,k)>0$ is valid uniformly for all $G$. This result is shown to hold more generally for any translation-invariant system of $r$ linear equations given by an integer matrix with coprime $r\times r$ minors.

math.CO↗

arXiv2015

On the removal lemma for linear configurations in finite abelian groups

We present a general framework to represent discrete configuration systems using hypergraphs. This representation allows one to transfer combinatorial removal lemmas to their analogues for configuration systems. These removal lemmas claim that a system without many configurations can be made configuration-free by removing a few of its constituent elements. As applications of this approach we give, an alternative proof of the removal lemma for permutations by Cooper, a general version of a removal lemma for linear systems in finite abelian groups, an interpretation of the mentioned removal lemma in terms of subgroups, and an alternative proof of the counting version of the multidimensional Szemerédi theorem in abelian groups with generalizations.

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arXiv2012

On the Removal Lemma for Linear Systems over Abelian Groups

In this paper we present an extension of the removal lemma to integer linear systems over abelian groups. We prove that, if the $k$--determinantal of an integer $(k\times m)$ matrix $A$ is coprime with the order $n$ of a group $G$ and the number of solutions of the system $Ax=b$ with $x_1\in X_1,..., x_m\in X_m$ is $o(n^{m-k})$, then we can eliminate $o(n)$ elements in each set to remove all these solutions. This is a follow-up of our former paper 'A Removal Lemma for Systems of Linear Equations over Finite Fields' arXiv:0809.1846v1, which dealt with the case of finite fields.

math.CO↗

arXiv2012

On the number of monochromatic solutions of integer linear systems on Abelian groups

Let $G$ be a finite abelian group with exponent $n$, and let $r$ be a positive integer. Let $A$ be a $k\times m$ matrix with integer entries. We show that if $A$ satisfies some natural conditions and $|G|$ is large enough then, for each $r$--coloring of $G\setminus \{0\}$, there is $δ$ depending only on $r,n$ and $m$ such that the homogeneous linear system $Ax=0$ has at least $δ|G|^{m-k}$ monochromatic solutions. Density versions of this counting result are also addressed.

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arXiv2008

A Removal Lemma for Systems of Linear Equations over Finite Fields

We prove a removal lemma for systems of linear equations over finite fields: let $X_1,...,X_m$ be subsets of the finite field $\F_q$ and let $A$ be a $(k\times m)$ matrix with coefficients in $\F_q$ and rank $k$; if the linear system $Ax=b$ has $o(q^{m-k})$ solutions with $x_i\in X_i$, then we can destroy all these solutions by deleting $o(q)$ elements from each $X_i$. This extends a result of Green [Geometric and Functional Analysis 15(2) (2005), 340--376] for a single linear equation in abelian groups to systems of linear equations. In particular, we also obtain an analogous result for systems of equations over integers, a result conjectured by Green. Our proof uses the colored version of the hypergraph Removal Lemma.

math.CO↗

arXiv2008

A combinatorial proof of the Removal Lemma for Groups

Green [Geometric and Functional Analysis 15 (2005), 340--376] established a version of the Szemerédi Regularity Lemma for abelian groups and derived the Removal Lemma for abelian groups as its corollary. We provide another proof of his Removal Lemma that allows us to extend its statement to all finite groups. We also discuss possible extensions of the Removal Lemma to systems of equations.

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