KAIST Discrete Math Seminar

We first present how to extend Ramanujan’s method in partition congruences and show a congruence relation that the coefficients of the quotient of generating series for partitions and sums of squares satisfy. Then we observe a combinatorial interpretation of the product of them and see whether we could find some arithmetic properties of its coefficients.

In their 1984 book “Algebraic Combinatorics I: Association Schemes”, E. Bannai and T. Ito conjectured that there are only finitely many distance-regular graphs with fixed valency k≥3.

In the series of papers, they showed that their conjecture holds for k=3, 4, and for the class of bipartite distance-regular graphs. J. H. Koolen and V. Moulton also show that there are only finitely many distance-regular graphs with k=5, 6, or 7, and there are only finitely many triangle-free distance-regular graphs with k=8, 9 or 10. In this talk, we show that the Bannai-Ito conjecture holds for any integer k>2 (i.e., for fixed integer k>2, there are only finitely many distance-regular graphs with valency k).

This is a joint work with A. Dubickas, J. H. Koolen and V. Moulton.

Lovász and Plummer conjectured that there exists a fixed positive constant c such that every cubic n-vertex graph with no cutedge has at least 2^cn perfect matchings. Their conjecture has been verified for bipartite graphs by Voorhoeve and planar graphs by Chudnovsky and Seymour. We prove that every claw-free cubic n-vertex graph with no cutedge has more than 2^n/18 perfect matchings, thus verifying the conjecture for claw-free graphs.

The graph finding problem is to find the edges of an unknown graph by using a certain type of queries. Its extension to hypergraphs is closely related to the problem of learning linkage in molecular biology and artificial intelligence. In this talk, we introduce the hypergraph finding problem and the linkage learning problem and present our recent results for the query complexity of those problems.