KAIST Discrete Math Seminar

The square of a graph is the graph defined on such that two vertices and are adjacent in if the distance between and in is at most 2. Let and be the chromatic number and the list chromatic number of , respectively. A graph is called if . It is an interesting problem to find graphs that are chromatic-choosable.</p>

Motivated by the List Total Coloring Conjecture, Kostochka and Woodall (2001) proposed the List Square Coloring Conjecture which states that is chromatic-choosable for every graph . Recently, Kim and Park showed that the List Square Coloring Conjecture does not hold in general by finding a family of graphs whose squares are complete multipartite graphs with partite sets of unbounded size. It is a well-known fact that the List Total Coloring Conjecture is true if the List Square Coloring Conjecture holds for special class of bipartite graphs. On the other hand, the counterexamples to the List Square Coloring Conjecture are not bipartite graphs. Hence a natural question is whether is chromatic-choosable or not for every bipartite graph .

In this paper, we give a bipartite graph such that . Moreover, we show that the value can be arbitrarily large. This is joint work with Boram Park.

Graph immersion is a natural containment relation like graph minors. However, until recently, graph immersion has received relatively little attention. In this talk we shall describe some recent progress toward understanding when a graph does not immerse a certain subgraph. Namely, we detail a rough structure theorem for graphs which do not have K_t as an immersion, and we discuss the precise structure of graphs which do not have K_{3,3} as an immersion.</p>

Then we turn our attention to a special class of digraphs, those for which every vertex has both indegree and outdegree equal to 2. These digraphs have special embeddings in surfaces where every vertex has a local rotation in which the inward and outward edges alternate. It turns out that the nature of these embeddings relative to immersion is quite closely related to the usual theory of graph embedding and graph minors. Here we describe the complete list of forbidden immersions for (special) embeddings in the projective plane.

These results are joint with various coauthors including Archdeacon, Dvorak, Fox, Hannie, Malekian, McDonald, Mohar, and Scheide.

Sidorenko’s conjecture states that for every bipartite graph on
\begin{eqnarray*}
\int \prod_{(i,j)\in E(H)} h(x_i, y_j) d\mu^{|V(H)|}
\ge \left( \int h(x,y) \,d\mu^2 \right)^{|E(H)|}
\end{eqnarray*}
holds, where is the Lebesgue measure on and is a bounded, non-negative, symmetric, measurable function on . An equivalent discrete form of the conjecture is that the number of homomorphisms from a bipartite graph to a graph is asymptotically at least the expected number of homomorphisms from to the Erdos-Renyi random graph with the same expected edge density as . In this talk, we will give an overview on known results and new approaches to attack Sidorenko’s conjecture.
This is a joint work with Jeong Han Kim and Choongbum Lee.

I intend to give an introduction to some of the wonderful topics in the world of additive combinatorics. This is a broad subject which features numerous different tools and techniques, and is presently a hotbed of exciting research. My focus will be on the combinatorics, and I will keep things as basic as possible (I will assume nothing more than a basic background in combinatorics). I’ll begin the tour with some of the classical theorems like Cauchy-Davenport and Erdos-Ginzburg-Ziv and I will exhibit some very clean proofs of these and other results such as the Theorems of Schrijver-Seymour, Green-Ruzsa, Dvir, and Elekes. We will also discuss (but not prove) some more recent results like the Breulliard-Green-Tao Theorem.

A Halin graph is constructed from a plane embedding of a tree whose non-leaf vertices have degree at least 3 by adding a cycle through its leaves in the natural order determined by the embedding. In this talk, we prove that every 3-connected -free graph has a spanning Halin subgraph. This result is best possible in the sense that the statement fails if is replaced by or is replaced by . This is a joint work with Guantao Chen, Jie Han, Songling Shan, and Shoichi Tsuchiya.