기초과학연구원 이산수학그룹
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 …
We introduce delta-graphic matroids, which are matroids whose bases form graphic delta-matroids. The class of delta-graphic matroids contains graphic matroids as well as cographic matroids and is a proper subclass of the class of regular matroids. We give a structural characterization of the class of delta-graphic matroids. We also show that every forbidden minor for …
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 …
Ramsey's theorem states that, for a fixed graph $H$, every 2-edge-colouring of $K_n$ contains a monochromatic copy of $H$ whenever $n$ is large enough. Perhaps one of the most natural questions after Ramsey's theorem is then how many copies of monochromatic $H$ can be guaranteed to exist. To formalise this question, let the Ramsey multiplicity …
There has been much research on finding a large rainbow matching in a properly edge-colored graph, where a proper edge coloring is a coloring of the edge set such that no same-colored edges are incident. Barát, Gyárfás, and Sárközy conjectured that in every proper edge coloring of a multigraph (with parallel edges allowed, but not …
A graph $H$ is common if the number of monochromatic copies of $H$ in a 2-edge-colouring of the complete graph $K_n$ is minimised by the random colouring. Burr and Rosta, extending a famous conjecture by Erdős, conjectured that every graph is common. The conjectures by Erdős and by Burr and Rosta were disproved by Thomason and …
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 …
I will go over the history on the study of the set of cycle lengths of graphs with large average degree or chromatic number, and discuss recent work with Richard Montgomery on this topic. In particular, we will see the divergence of harmonic sum of odd cycle lengths in graphs with large chromatic number and …
In this talk I will state a generalisation of the even directed cycle problem, which asks whether a given digraph contains a directed cycle of even length, to orientations of regular matroids. Motivated by this problem, I will define non-even oriented matroids generalising non-even digraphs, which played a central role in resolving the computational complexity of …
In combinatorics, Hopf algebras appear naturally when studying various classes of combinatorial objects, such as graphs, matroids, posets or symmetric functions. Given such a class of combinatorial objects, basic information on these objects regarding assembly and disassembly operations are encoded in the algebraic structure of a Hopf algebra. One then hopes to use algebraic identities of …