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

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Recent Submissions

  • Structure and enumeration of K4-minor-free links and link-diagrams 

    Rué Perna, Juan José; Thilikos Touloupas, Dimitrios; Velona, Vasiliki (2020-10-01)
    Article
    Open Access
    We study the class of link-types that admit a -minor-free diagram, i.e., they can be projected on the plane so that the resulting graph does not contain any subdivision of . We prove that is the closure of a subclass of ...
  • On the number of coloured triangulations of d-manifolds 

    Chapuy, G.; Perarnau Llobet, Guillem (2020-03-11)
    Article
    Open Access
    We give superexponential lower and upper bounds on the number of coloured d-dimensional triangulations whose underlying space is an oriented manifold, when the number of simplices goes to infinity and d=3 is fixed. In the ...
  • On varieties defined by large sets of quadrics and their application to error-correcting codes 

    Ball, Simeon Michael; Pepe, Valentina (2020-10-01)
    Article
    Restricted access - publisher's policy
    Let U be a ( k-1 2 - 1)-dimensional subspace of quadratic forms defined on F k with the property that U does not contain any reducible quadratic form. Let V (U) be the points of PG(k - 1, F) which are zeros of all quadratic ...
  • The Glauber dynamics for edge-colorings of trees 

    Delcourt, Michelle; Heinrich, Marc; Perarnau Llobet, Guillem (2020-09-26)
    Article
    Restricted access - publisher's policy
    Let T be a tree on n vertices and with maximum degree ¿. We show that for k = ¿ + 1 the Glauber dynamics for k-edge-colourings of T mixes in polynomial time in n. The bound on the number of colours is best possible as the ...
  • Sidon set systems 

    Cilleruelo, Javier; Serra Albó, Oriol; Wötzel, Maximilian (2020-02-11)
    Article
    Open Access
    A family A of k-subsets of {1,2,…,N} is a Sidon system if the sumsets A+B, A,B¿A are pairwise distinct. We show that the largest cardinality Fk(N) of a Sidon system of k-subsets of [N] satisfies Fk(N)=(N-1k-1)+N-k and the ...
  • A step beyond Freiman’s theorem for set addition modulo a prime 

    Candela Pokorna, Pablo; Serra Albó, Oriol; Spiegel, Christoph (2020-01-01)
    Article
    Open Access
    Freiman’s 2.4-Theorem states that any set A¿Zp satisfying |2A|=2.4|A|-3 and |A|<p/35 can be covered by an arithmetic progression of length at most |2A|-|A|+1. A more general result of Green and Ruzsa implies that this ...
  • On the smallest trees with the same restricted U-polynomial and the rooted U-polynomial 

    Aliste Prieto, José; Mier Vinué, Anna de; Zamora Ponce, José (2021-03-01)
    Article
    Open Access
    In this article, we construct explicit examples of pairs of non-isomorphic trees with the same restricted U -polynomial for every k; by this we mean that the polynomials agree on terms with degree at most k + 1. The main ...
  • Diameter and stationary distribution of random r-out digraphs 

    Addario-Berry, Louigi; Balle, Borja; Perarnau Llobet, Guillem (2020-08-07)
    Article
    Open Access
    Let D(n, r) be a random r-out regular directed multigraph on the set of vertices {1, . . . , n}. In this work, we establish that for every r = 2, there exists ¿r > 0 such that diam(D(n, r)) = (1 + ¿r + o(1)) logr n. The ...
  • On list k-coloring convex bipartite graphs 

    Díaz Cort, Josep; Yasar Diner, Oznur; Serna Iglesias, María José; Serra Albó, Oriol (Springer, 2020)
    Conference report
    Restricted access - publisher's policy
    List k–Coloring (LI k-COL) is the decision problem asking if a given graph admits a proper coloring compatible with a given list assignment to its vertices with colors in {1,2,..., k}. The problem is known to be NP-hard ...
  • A rainbow Dirac's theorem 

    Coulson, Matthew John; Perarnau Llobet, Guillem (2020-01-01)
    Article
    Open Access
    A famous theorem of Dirac states that any graph on n vertices with minimum degree at least n/2 has a Hamilton cycle. Such graphs are called Dirac graphs. Strengthening this result, we show the existence of rainbow Hamilton ...
  • Enumeration of labelled 4-regular planar graphs 

    Noy Serrano, Marcos; Requilé, Clément; Rué Perna, Juan José (2019-08-01)
    Article
    Open Access
    We present the first combinatorial scheme for counting labelled 4-regular planar graphs through a complete recursive decomposition. More precisely, we show that the exponential generating function of labelled 4-regular ...
  • Further results on random cubic planar graphs 

    Noy Serrano, Marcos; Requilé, Clément; Rué Perna, Juan José (2020-06)
    Article
    Restricted access - publisher's policy
    We provide precise asymptotic estimates for the number of several classes of labeled cubic planar graphs, and we analyze properties of such random graphs under the uniform distribution. This model was first analyzed by ...

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