기초과학연구원 이산수학그룹
In 2017, Aharoni proposed the following generalization of the Caccetta-Häggkvist conjecture for digraphs. If G is a simple n-vertex edge-colored graph with n color classes of size at least r, then G contains a rainbow cycle of length at most ⌈n/r⌉. In this talk, we prove that Aharoni's conjecture holds up to an additive constant. …
A cross-cap drawing of a graph G is a drawing on the sphere with g distinct points, called cross-caps, such that the drawing is an embedding except at the cross-caps, where edges cross properly. A cross-cap drawing of a graph G with g cross-caps can be used to represent an embedding of G on a …
Very little is known about critical properties of graphs in the hierarchy of monotone classes, i.e. classes closed under taking (not necessarily induced) subgraphs. We distinguish four important levels in this hierarchy and discuss possible new levels by focusing on the Hamiltonian cycle problem. In particular, we obtain a number of results for this problem …
Two-way online correlated selection (two-way OCS) is an online algorithm that, at each timestep, takes a pair of elements from the ground set and irrevocably chooses one of the two elements, while ensuring negative correlation in the algorithm's choices. OCS was initially invented by Fahrbach, Huang, Tao, and Zadimoghaddam (FOCS 2020, JACM 2022) to break …
A vertex colouring of a hypergraph is $c$-strong if every edge $e$ sees at least $\min\{c, |e|\}$ distinct colours. Let $\chi(t,c)$ denote the least number of colours needed so that every $t$-intersecting hypergraph has a $c$-strong colouring. In 2012, Blais, Weinstein and Yoshida introduced this parameter and initiated study on when $\chi(t,c)$ is finite: they …
We will discuss recent progress on the topic of induced subgraphs and tree-decompositions. In particular this talk with focus on the proof of a conjecture of Hajebi that asserts that (if we exclude a few obvious counterexamples) for every integer t, every graph with large enough treewidth contains two anticomplete induced subgraphs each of treewidth …
We provide new constructions of families of quasi-random graphs that behave like Paley graphs but are neither Cayley graphs nor Cayley sum graphs. These graphs give a unified perspective of studying various graphs defined by polynomials over finite fields, such as Paley graphs, Paley sum graphs, and graphs associated with Diophantine tuples and their generalizations …
Since the introduction of cluster algebras by Fomin and Zelevinsky in 2002, there has been significant interest in cluster algebras of surface type. These algebras are particularly noteworthy due to their ability to construct various combinatorial structures like snake graphs, T-paths, and posets, which are useful for proving key structural properties such as positivity or …
Tropical geometry replaces usual addition and multiplication with tropical addition (the min) and tropical multiplication (the sum), which offers a polyhedral interpretation of algebraic variety. This talk aims to pitch the usefulness of tropical geometry in understanding classical algebraic geometry. As an example, we introduce the tropicalization of the variety of symmetric rank 2 matrices. …
For a given hypergraph $H$ and a vertex $v\in V(H)$, consider a random matching $M$ chosen uniformly from the set of all matchings in $H.$ In $1995,$ Kahn conjectured that if $H$ is a $d$-regular linear $k$-uniform hypergraph, the probability that $M$ does not cover $v$ is $(1 + o_d(1))d^{-1/k}$ for all vertices $v\in V(H)$. …