GAPCOMB - Geometric, Algebraic and Probabilistic Combinatorics

 

The research of the group focuses on interrelated aspects of combinatorics: Graph theory, Random Graphs, Probabilistic method, Geometric group theory and algebraic methods, Enumerative combinatorics, Combinatorial geometry, and Combinatorial number theory. Some recent research achievements are: Proof of the Maximum Distance Separable codes conjecture for prime fields; solution of a conjecture of Green on the removal lemma for systems of equations in finite fields

Recent Submissions

  • A telescopic proof of Cayley’s formula 

    Chapuy, Guillaume; Perarnau Llobet, Guillem (2024-09-24)
    Article
    Restricted access - publisher's policy
    We give a short proof of the fact that the number of labeled trees on n vertices is n n - 2 . Although many short proofs are known, we have not seen this one before.

  • Orbit problems for free-abelian times free groups and related families 

    Carvalho, André; Delgado Rodríguez, Jordi (2024-11-07)
    Article
    Restricted access - publisher's policy
    We prove that the Brinkmann Problems (BrP & BrCP) and the Twisted-Conjugacy Problem (TCP) are decidable for any endomorphism of a free-abelian times free (FATF) group F n × Z m . Furthermore, we prove the decidability of ...

  • Grafs aleatoris amb una seqüència de graus donada 

    Perarnau Llobet, Guillem (Societat Catalana de Matemàtiques (SCM), 2023)
    Article
    Open Access
    Des de l’aparició de la noció de xarxes complexes, els grafs aleatoris han resultat una eina fonamental per modelar-les i estudiar-les. En aquest article tractarem l’estudi dels grafs aleatoris amb una seqüència de graus ...

  • Product-free sets in the free group 

    Ortega Sánchez Colomer, Miquel; Rué Perna, Juan José; Serra Albó, Oriol (2024-07)
    Article
    Open Access
    We prove that product-free subsets of the free group over a finite alphabet have maximum upper density mathematical equation with respect to the natural measure that assigns total weight one to each set of irreducible words ...

  • On homogeneous matroid ports 

    Martí Farré, Jaume; Mier Vinué, Anna de (Universidad de Alcalá, 2024)
    Conference lecture
    Open Access
    A matroid port is a clutter (antichain of sets) determined by the collection of circuits of a matroid that contain a fixed point. We study the problem of determining matroid ports all whose elements have the same size h. ...

  • Decidability of the Brinkmann problems for endomorphisms of the free group 

    Carvalho, André; Delgado Rodríguez, Jordi (Springer, 2024-07-18)
    Article
    Restricted access - publisher's policy
    Building on the work of Brinkmann and Logan, we show that both the Brinkmann problem and the Brinkmann conjugacy problem are decidable for endomorphisms of the free group Fn.

  • Enhancing mathematical research in high school 

    Morera, Laura; Rué Perna, Juan José; Alet, Ferran (Springer, 2012)
    Conference lecture
    Open Access
    We present a didactic experience in the context of enhance mathematics research in high school. The encounter of three different profiles (a skilled student, a mathematics school teacher and a researcher in mathematics) ...

  • Strong Ramsey games: drawing on an infinite board 

    Hefetz, Dan; Kusch, C.; Narins, Lothar; Pokrovskiy, Alexey; Requile, Clement; Sarid, Amir (2017-08)
    Article
    Open Access
    Consider the following strong Ramsey game. Two players take turns in claiming a previously unclaimed edge of the complete graph on n vertices until all edges have been claimed. The first player to build a copy of K_q (q ...

  • The central tree property and algorithmic problems on subgroups of free groups 

    Roy, Mallika; Ventura Capell, Enric; Weil, Pascal (De Gruyter, 2024-02-14)
    Article
    Open Access
    We study the average case complexity of the Uniform Membership Problem for subgroups of free groups, and we show that it is orders of magnitude smaller than the worst case complexity of the best known algorithms. This ...

  • On minimum vertex bisection of random d-regular graphs 

    Díaz Cort, Josep; Yasar Diner, Oznur; Serna Iglesias, María José; Serra Albó, Oriol (2024-09)
    Article
    Open Access
    Minimum vertex bisection is a graph partitioning problem in which the aim is to find a partition of the vertices into two equal parts that minimizes the number of vertices in one partition set that have a neighbor in the ...

  • Short synchronizing words for random automata 

    Chapuy, Guillaume; Perarnau Llobet, Guillem (Curran Associates, Inc., 2023)
    Conference report
    Open Access
    We prove that a uniformly random automaton with n states on a 2-letter alphabet has a synchronizing word of length with high probability (w.h.p.). That is to say, w.h.p. there exists a word ¿ of such length, and a state ...

  • Percolation on dense random graphs with given degrees 

    Lichev, Lyuben; Mitsche, Dieter; Perarnau Llobet, Guillem (Elsevier, 2024-03-26)
    Article
    Open Access
    In this paper, we study the order of the largest connected component of a random graph having two sources of randomness: first, the graph is chosen randomly from all graphs with a given degree sequence, and then bond ...

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