기초과학연구원 이산수학그룹
Maximum induced matching width, also known as mim-width, is a width parameter for graphs introduced by Vatshelle in 2012. This parameter can be defined over branch decompositions of a graph G, where the width of a vertex partition (X,Y) in G is the size of a maximum induced matching in the bipartite graph induced by …
Forbidden graph characterizations provide a convenient way of specifying graph classes, which often exhibit a rich combinatorial and algorithmic theory. A prime example in graph theory are classes of bounded tree-width, which are characterized as those classes that exclude some planar graph as a minor. Similarly, in model theory, classes of structures are characterized by …
In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\ge 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log t})$ and hence is $O(t\sqrt{\log t})$-colorable. Recently, Norin, Song and I showed that every graph with no $K_t$ minor is …
We study the Excluded Grid Theorem, a fundamental structural result in graph theory, that was proved by Robertson and Seymour in their seminal work on graph minors. The theorem states that there is a function $f$, such that for every integer $g > 0$, every graph of treewidth at least $f(g)$ contains the g×g-grid as a minor. For every …
The asymptotic dimension of metric spaces is an important notion in geometric group theory. The metric spaces considered in this talk are the ones whose underlying spaces are the vertex-sets of (edge-)weighted graphs and whose metrics are the distance functions in weighted graphs. A standard compactness argument shows that it suffices to consider the asymptotic …
For each integer $k\ge 2$, we determine a sharp bound on $\operatorname{mad}(G)$ such that $V(G)$ can be partitioned into sets $I$ and $F_k$, where $I$ is an independent set and $G$ is a forest in which each component has at most k vertices. For each $k$ we construct an infinite family of examples showing our result is best …
In this talk I will present a small result we achieved during a workshop in February this year. My coauthors on this are Marcin Pilipczuk, Paweł Komosa and Manuel Sorge. A bramble in an undirected graph $G$ is a family of connected subgraphs of $G$ such that for every two subgraphs $H_1$ and $H_2$ in the bramble either $V(H_1) \cap …
Shrub-depth is a graph invariant often considered as an extension of tree-depth to dense graphs. In this talk I will explain our recent proofs of two results about graphs of bounded shrub-depth. Every graph property definable in monadic-second order logic, e.g., 3-colorability, can be evaluated by Boolean circuits of constant depth and polynomial size, whose …
In extremal graph theory, a graph G is H-saturated if G does not contain a copy of H but adding any missing edge to G creates a copy of H. The saturation number, sat(n, H), is the minimum number of edges in an n-vertex H-saturated graph. This class of problems was first studied by Zykov …
The study of Hamiltonian graphs, i.e. finite graphs having a cycle that contains all vertices of the graph, is a central theme of finite graph theory. For infinite graphs such a definition cannot work, since cycles are finite. We shall debate possible concepts of Hamiltonicity for infinite graphs and eventually follow the topological approach by …