arXiv subjects

Lluis Vena

Publications and source records attributed to Lluis Vena.

11 recordsLinked to original sources

arXiv2023

Characterisation of the big Ramsey degrees of the generic partial order

As a result of 33 intercontinental Zoom calls, we characterise big Ramsey degrees of the generic partial order. This is an infinitary extension of the well known fact that finite partial orders endowed with linear extensions form a Ramsey class (this result was announced by Ne\v{s}et\v{r}il and R\"odl in 1984 with first published proof by Paoli, Trotter and Walker in 1985). Towards this, we refine earlier upper bounds obtained by Hubi\v{c}ka based on a new connection of big Ramsey degrees to the Carlson-Simpson theorem and we also introduce a new technique of giving lower bounds using an iterated application of the upper-bound theorem.

math.CO↗

arXiv2021

Exact big Ramsey degrees for finitely constrained binary free amalgamation classes

We characterize the big Ramsey degrees of free amalgamation classes in finite binary languages defined by finitely many forbidden irreducible substructures, thus refining the recent upper bounds given by Zucker. Using this characterization, we show that the Fra\"iss\'e limit of each such class admits a strong big Ramsey structure, implying that the automorphism group of the Fra\"iss\'e limit has a metrizable universal completion flow.

math.LO↗

arXiv2021

Big Ramsey degrees of the generic partial order

As a result of 33 intercontinental Zoom calls, we characterise big Ramsey degrees of the generic partial order in a similar way as Devlin characterised big Ramsey degrees of the generic linear order (the order of rationals).

math.CO↗

arXiv2020

Big Ramsey degrees of 3-uniform hypergraphs are finite

We prove that the universal homogeneous 3-uniform hypergraph has finite big Ramsey degrees. This is the first case where big Ramsey degrees are known to be finite for structures in a non-binary language. Our proof is based on the vector (or product) form of Milliken's Tree Theorem and demonstrates a general method to carry existing results on structures in binary relational languages to higher arities.

math.CO↗

arXiv2020

Coloring graphs by translates in the circle

The fractional and circular chromatic numbers are the two most studied non-integral refinements of the chromatic number of a graph. Starting from the definition of a coloring base of a graph, which originated in work related to ergodic theory, we formalize the notion of a gyrocoloring of a graph: the vertices are colored by translates of a single Borel set in the circle group, and neighbouring vertices receive disjoint translates. The corresponding gyrochromatic number of a graph always lies between the fractional chromatic number and the circular chromatic number. We investigate basic properties of gyrocolorings. In particular, we construct examples of graphs whose gyrochromatic number is strictly between the fractional chromatic number and the circular chromatic number. We also establish several equivalent definitions of the gyrochromatic number, including a version involving all finite abelian groups.

math.CO↗

arXiv2019

Big Ramsey degrees of 3-uniform hypergraphs

Given a countably infinite hypergraph $\mathcal R$ and a finite hypergraph $\mathcal A$, the big Ramsey degree of $\mathcal A$ in $\mathcal R$ is the least number $L$ such that, for every finite $k$ and every $k$-colouring of the embeddings of $\mathcal A$ to $\mathcal R$, there exists an embedding $f$ from $\mathcal R$ to $\mathcal R$ such that all the embeddings of $\mathcal A$ to the image $f(\mathcal R)$ have at most $L$ different colours. We describe the big Ramsey degrees of the random countably infinite 3-uniform hypergraph, thereby solving a question of Sauer. We also give a new presentation of the results of Devlin and Sauer on, respectively, big Ramsey degrees of the order of the rationals and the countably infinite random graph. Our techniques generalise (in a natural way) to relational structures and give new examples of Ramsey structures (a concept recently introduced by Zucker with applications to topological dynamics).

math.CO↗

arXiv2019

Tutte's dichromate for signed graphs

We introduce the ``trivariate Tutte polynomial" of a signed graph as an invariant of signed graphs up to vertex switching that contains among its evaluations the number of proper colorings and the number of nowhere-zero flows. In this, it parallels the Tutte polynomial of a graph, which contains the chromatic polynomial and flow polynomial as specializations. The number of nowhere-zero tensions (for signed graphs they are not simply related to proper colorings as they are for graphs) is given in terms of evaluations of the trivariate Tutte polynomial at two distinct points. Interestingly, the bivariate dichromatic polynomial of a biased graph, shown by Zaslavsky to share many similar properties with the Tutte polynomial of a graph, does not in general yield the number of nowhere-zero flows of a signed graph. Therefore the ``dichromate" for signed graphs (our trivariate Tutte polynomial) differs from the dichromatic polynomial (the rank-size generating function). The trivariate Tutte polynomial of a signed graph can be extended to an invariant of ordered pairs of matroids on a common ground set -- for a signed graph, the cycle matroid of its underlying graph and its frame matroid form the relevant pair of matroids. This invariant is the canonically defined Tutte polynomial of matroid pairs on a common ground set in the sense of a recent paper of Krajewski, Moffatt and Tanasa, and was first studied by Welsh and Kayibi as a four-variable linking polynomial of a matroid pair on a common ground set.

math.CO↗

arXiv2018

A topological characterization of Gauss codes

A (smooth) embedding of a closed curve on the plane with finitely many intersections is said to be generic if each point of self-intersection is crossed exactly twice and at non-tangent angles. A finite word $\omega$ where each character occurs twice is a Gauss code if it can be obtained as the sequence of traversed self-intersections of a generic plane embedding of a close curved $\gamma$. Then $\gamma$ is said to realize $\omega$. We present a characterization of Gauss codes using Seifert cycles. The characterization is given by an algorithm that, given a Gauss code $\omega$ as input, it outputs a combinatorial plane embedding of closed curve in linear time with respect to the number of characters of the word. The algorithm allows to find all the (combinatorial) embeddings of a closed curve in the plane that realize $\omega$. The characterization involve two functions between graphs embedded on orientable surfaces, invertible of each other. One produces an embedding of the Seifert graph of the word (or paragraph), and the other is its inverse operation. These operations might be of independent interest.

math.CO↗

arXiv2018

On the complexity of solving a decision problem with flow-depending costs: the case of the IJsselmeer dikes

We consider a fundamental integer programming (IP) model for cost-benefit analysis flood protection through dike building in the Netherlands, due to Verweij and Zwaneveld. Experimental analysis with data for the Ijsselmeer lead to integral optimal solution of the linear programming relaxation of the IP model. This naturally led to the question of integrality of the polytope associated with the IP model. In this paper we first give a negative answer to this question by establishing non-integrality of the polytope. Second, we establish natural conditions that guarantee the linear programming relaxation of the IP model to be integral. We then test the most recent data on flood probabilities, damage and investment costs of the IJsselmeer for these conditions. Third, we show that the IP model can be solved in polynomial time when the number of dike segments, or the number of feasible barrier heights, are constant.

q-fin.RM↗

arXiv2016

A Tutte polynomial for maps

We follow the example of Tutte in his construction of the dichromate of a graph (that is, the Tutte polynomial) as a unification of the chromatic polynomial and the flow polynomial in order to construct a new polynomial invariant of maps (graphs embedded in orientable surfaces). We call this the surface Tutte polynomial. The surface Tutte polynomial of a map contains the Las Vergnas polynomial, Bollob\'as-Riordan polynomial and Kruskhal polynomial as specializations. By construction, the surface Tutte polynomial includes among its evaluations the number of local tensions and local flows taking values in any given finite group. Other evaluations include the number of quasi-forests.

math.CO↗

arXiv2015

Matroid invariants and counting graph homomorphisms

The number of homomorphisms from a finite graph $F$ to the complete graph $K_n$ is the evaluation of the chromatic polynomial of $F$ at $n$. Suitably scaled, this is the Tutte polynomial evaluation $T(F;1-n,0)$ and an invariant of the cycle matroid of $F$. De la Harpe and Jaeger \cite{dlHJ95} asked more generally when is it the case that a graph parameter obtained from counting homomorphisms from $F$ to a fixed graph $G$ depends only on the cycle matroid of $F$. They showed that this is true when $G$ has a generously transitive automorphism group (examples include Cayley graphs on an abelian group, and Kneser graphs). Using tools from multilinear algebra, we prove the converse statement, thus characterizing finite graphs $G$ for which counting homomorphisms to $G$ yields a matroid invariant. We also extend this result to finite weighted graphs $G$ (where to count homomorphisms from $F$ to $G$ includes such problems as counting nowhere-zero flows of $F$ and evaluating the partition function of an interaction model on $F$).

math.CO↗