KAIST Discrete Math Seminar

Given a graph, we are interested in studying the problem of finding an induced subgraph of a fixed order with largest number of edges. More generally, let G = (V, E) be an undirected graph, with a weight (budget) function on the vertices, w: V → ℤ⁺, and a benefit function on vertices and edges b: E ∪ V → ℤ. The benefit of a subgraph H =(V_H,E_H) is b(H) = ∑ _v∈VH b(v) + ∑ _e∈EH b(e) while its weight is w(H) = ∑ _v∈VH w(v). What can be said about the maximum benefit of an induced subgraph with the restriction that its weight is less than W?

This problem is closely related to the Quadratic Knapsack Problem, the Densest Subgraph Problem, and classical problems in Extremal Graph Theory. We will discuss these connections, give applications in resource allocation, and present new results on approximation algorithms using methods from convex optimization and probability. This is joint work with Kapoor.

A small cover is a topological analogue of real toric varieties, and is an important object in toric topology. It is noted that the formula of the ℤ₂-cohomology ring of small cover is well-known. However, the integral cohomology ring of small covers has not been known well.

In this talk, we discuss about the Betti numbers and its torsion of the small covers associated to some nestohedra including graph associahedra. Interestingly, the Betti numbers can be computed by purely combinatorial method (in terms of graphs and hypergraphs). To our surprise, for specific families of graphs, these numbers are deeply related to well-known combinatorial sequences such as the Catalan numbers and Euler zigzag numbers.

Given a simple polygon P, a point t in P, and a set of positive weights, we want to place the weights on the boundary of P in such a way that their barycenter comes to t. We show that there is always such a placement if the weights are balanced, i.e., no weight is larger than half of the total weight, and the placement can be found efficiently. We also study three-dimensional versions of the problem.</p>

Joint work with Luis Barba, Jean Lou De Carufel, Rudolf Fleischer, Akitoshi Kawamura, Matias Korman, Yuan Tang, Takeshi Tokuyama, Sander Verdonschot, and Tianhao Wang.

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 ⟨J_λ[X;q,q^k]/(1-q)ⁿ,s_μ(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.

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 χ_l(H) be the chromatic number and the list chromatic number of H, respectively. A graph H is called chromatic-choosable if χ_l(H) = χ(H). It is an interesting problem to find graphs that are chromatic-choosable. Kostochka and Woodall conjectured that χ_l(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 χ_l(G²) − χ(G²) can be arbitrary large.