Seminars in November 2019

  • Frédéric Meunier, Topological bounds for graph representations over any field

    IBS/KAIST Joint Discrete Math Seminar

    Topological bounds for graph representations over any field
    Frédéric Meunier
    École Nationale des Ponts et Chaussées, Paris
    2019/11/21 Thu 4:30PM-5:30PM, Room B232, IBS (기초과학연구원)
    Haviv (European Journal of Combinatorics, 2019) has recently proved that some topological lower bounds on the chromatic number of graphs are also lower bounds on their orthogonality dimension over R. We show that this holds actually for all known topological lower bounds and all fields. We also improve the topological bound he obtained for the minrank parameter over R – an important graph invariant from coding theory – and show that this bound is actually valid for all fields as well. The notion of independent representation over a matroid is introduced and used in a general theorem having these results as corollaries. Related complexity results are also discussed. This is joint work with Meysam Alishahi.

  • Ruth Luo, Induced Turán problems for hypergraphs

    IBS/KAIST Joint Discrete Math Seminar

    Induced Turán problems for hypergraphs
    Ruth Luo
    University of California, San Diego
    2019/11/19 Tue 4:30PM-5:30PM (Room B232, IBS)
    Let $F$ be a graph. We say that a hypergraph $\mathcal H$ is an induced Berge $F$ if there exists a bijective mapping $f$ from the edges of $F$ to the hyperedges of $\mathcal H$ such that for all $xy \in E(F)$, $f(xy) \cap V(F) = \{x,y\}$. In this talk, we show asymptotics for the maximum number of edges in $r$-uniform hypergraphs with no induced Berge $F$. In particular, this function is strongly related to the generalized Turán function $ex(n,K_r, F)$, i.e., the maximum number of cliques of size $r$ in $n$-vertex, $F$-free graphs.  Joint work with Zoltan Füredi.
    Tags:

  • Tony Huynh, Stable sets in graphs with bounded odd cycle packing number

    IBS/KAIST Joint Discrete Math Seminar

    Stable sets in graphs with bounded odd cycle packing number
    Tony Huynh
    Monash University
    2019/11/12 Tue 4:30PM-5:30PM (Room B232, IBS)
    It is a classic result that the maximum weight stable set problem is efficiently solvable for bipartite graphs.  The recent bimodular algorithm of Artmann, Weismantel and Zenklusen shows that it is also efficiently solvable for graphs without two disjoint odd cycles.  The complexity of the stable set problem for graphs without $k$ disjoint odd cycles is a long-standing open problem for all other values of $k$.  We prove that under the additional assumption that the input graph is embedded in a surface of bounded genus, there is a polynomial-time algorithm for each fixed $k$.  Moreover, we obtain polynomial-size extended formulations for the respective stable set polytopes. To this end, we show that 2-sided odd cycles satisfy the Erdős-Pósa property in graphs embedded in a fixed surface. This result may be of independent interest and extends a theorem of Kawarabayashi and Nakamoto asserting that odd cycles satisfy the Erdős-Pósa property in graphs embedded in a fixed orientable surface. Eventually, our findings allow us to reduce the original problem to the problem of finding a minimum-cost non-negative integer circulation of a certain homology class, which we prove to be efficiently solvable in our case. This is joint work with Michele Conforti, Samuel Fiorini, Gwenaël Joret, and Stefan Weltge.
    Tags:

  • Sun Kim (김선), Two identities in Ramanujan’s Lost Notebook with Bessel function series
    Two identities in Ramanujan’s Lost Notebook with Bessel function series
    Sun Kim (김선)
    Mathematical Institute, University of Cologne, Cologne, Germany
    2019/11/05 Tue 4:30PM-5:30PM (room 1401, E6-1, KAIST)
    On page 335 in his lost notebook, Ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of Bessel functions. We proved each of these identities under three different interpretations for the double series, and showed that they are intimately connected with the classical circle and divisor problems in number theory. Furthermore, we established many analogues and generalizations of them. This is joint work with Bruce C. Berndt and Alexandru Zaharescu.
    Tags:

Monthly Archives