Seminars in October 2013

  • Meesue Yoo (유미수), Schur expansion of the integral form of Macdonald polynomials
    Schur expansion of the integral form of Macdonald polynomials
    Meesue Yoo (유미수)
    KIAS
    2013/10/30 Wednesday 4PM-5PM
    ROOM 1409
    In this talk, we consider the combinatorial formula for the Schur coefficients of the integral form of the Macdonald polynomials. As an attempt to prove Haglund’s conjecture that ⟨Jλ[X;q,qk]/(1-q)n,sμ(X)⟩∈ℕ[q], we have found explicit combinatorial formulas for the Schur coefficients in one row case, two column case and certain hook shape cases. Egge-Loehr-Warrington constructed a combinatorial way of getting Schur expansion of symmetric functions when the expansion of the function in terms of Gessel’s fundamental quasi symmetric functions is given. We apply this method to the combinatorial formula for the integral form Macdonlad polynomials of Haglund-Haiman-Loehr in quasi symmetric functions to get the Schur coefficients and prove the Haglund’s conjecture in more general cases.
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  • Boram Park (박보람), Counterexamples to the List Square Coloring Conjecture
    Counterexamples to the List Square Coloring Conjecture
    Boram Park
    Optimization and Its Application Research Team
    NIMS
    2013/10/16 Wednesday 4PM-5PM
    ROOM 1409
    The square G2 of a graph G is the graph defined on V(G) such that two vertices u and v are adjacent in G2 if the distance between u and v in G is at most 2. Let χ(H) and χl(H) be the chromatic number and the list chromatic number of H, respectively. A graph H is called chromatic-choosable if χl(H) = χ(H). It is an interesting problem to find graphs that are chromatic-choosable. Kostochka and Woodall conjectured that χl(G2) = χ(G2) for every graph G, which is called List Square Coloring Conjecture. In this paper, we give infinitely many counterexamples to the conjecture. Moreover, we show that the value χl(G2) − χ(G2) can be arbitrary large.
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  • O-Joung Kwon (권오정), Unavoidable vertex-minors in large prime graphs. (FRIDAY)
    Unavoidable vertex-minors in large prime graphs
    O-joung Kwon
    Department of Mathematical Sciences
    KAIST
    2013/10/04 Friday 4PM-5PM
    ROOM 1409
    A split of a graph is a partition (A,B) of the vertex set V(G) having subsets A of A and B of B such that |A|,|B| > 1 and a vertex a in A is adjacent to a vertex b in B if and only if a is in A and b is in B. A graph is prime (with respect to the split decomposition) if it has no split.</p>

    We prove that for each n, there exists N such that every prime graph on at least N vertices contains a vertex-minor isomorphic to either a cycle of length n or a graph consisting of two disjoint cliques of size n joined by a matching.

    In this talk, we plan to describe a main tool, which is called a blocking sequence in a prime graph, and we will describe two big steps of the proof. And we will pose some open problems behind this result.

    This is a joint work with Sang-il Oum.

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