Let $G = (V, E)$ be a graph on $n$ vertices, and let $c : E \to P$, where $P$ is a set of colors. Let $\delta^c(G) = \min_{v \in V} \{ d^{c}(v) \}$ where $d^c(v)$ is the number of colors on edges incident to a vertex $v$ of $G$. In 2011, Fujita and Magnant showed that if $G$ is a graph on $n$ vertices that satisfies $\delta^c(G)\geq n/2$, then for every two vertices $u, v$ there is a properly-colored $u,v$-path in $G$. We show that for sufficiently large graphs $G$, the same bound for $\delta^c(G)$ implies that any two vertices are connected by a rainbow path. We also show sufficient conditions on $\delta^c(G)$ for the existence of a rainbow cycle of length $2k$ in sufficiently large bipartite graphs $G$, which are tight in many cases. This is joint work with Andrzej Czygrinow.

Flag algebras are a mathematical framework introduced by Alexander Razborov in 2007, which has been used to resolve a wide range of open problems in extremal graph theory in the past twenty years. This framework provides an algebraic setup where one can express relationships between induced subgraph densities symbolically. It also comes with mathematical techniques for systematically deriving such relationships that always hold. Some of these techniques have even been implemented in automatic tools, such as Flagmatic. In this work, we formalise flag algebras in Lean, an interactive theorem prover based on dependent type theory. Lean is computer software that lets us write mathematical definitions and proofs in a computer and check the correctness of written proofs using a computer. By formalizing flag algebras in Lean, we can ensure that the mathematical foundation of flag algebras does not have any mistakes, and provide a mathematical guarantee that theorems proved by flag algebras are indeed correct. In this talk, I will explain flag algebras and our Lean formalization using Mantel’s theorem as a guiding example. In the process, I will describe the challenges and lessons learned during the formalization. This is a joint work with Jihoon Hyun, Gyeongwon Jeong, Sang-il Oum, and Hongseok Yang.

The planar separator theorem by Lipton and Tarjan [FOCS ’77, SIAM Journal on Applied Mathematics ’79] states that any planar graph with $n$ vertices has a balanced separator of size $O(\sqrt{n})$ that can be found in linear time. This landmark result kicked off decades of research on designing linear or nearly linear-time algorithms on planar graphs. In an attempt to generalize Lipton-Tarjan’s theorem to nonplanar graphs, Alon, Seymour, and Thomas [STOC ’90, Journal of the AMS ’90] showed that any minor-free graph admits a balanced separator of size $O(\sqrt{n})$ that can be found in $O(n^{3/2})$ time. The superlinear running time in their separator theorem is a key bottleneck for generalizing algorithmic results from planar to minor-free graphs. Despite extensive research for more than two decades, finding a balanced separator of size $O(\sqrt{n})$ in (linear) $O(n)$ time for minor-free graphs remains a major open problem. Known algorithms either give a separator of size much larger than $O(\sqrt{n})$ or have superlinear running time, or both. In this paper, we answer the open problem affirmatively. Our algorithm is very simple: it runs a vertex-weighted variant of breadth-first search (BFS) a constant number of times on the input graph. Our key technical contribution is a weighting scheme on the vertices to guide the search for a balanced separator, offering a new connection between the size of a balanced separator and the existence of a clique-minor model. We believe that our weighting scheme may be of independent interest. This is a joint work with Édouard Bonnet, Tuukka Korhonen, Hung Le, and Jason Li.

Boolean function analysis for the hypercube $\{ 0, 1 \}^n$ is a well-developed field and has many famous results such as the FKN Theorem or Nisan-Szegedy Theorem. One easy example is the classification of Boolean degree $1$ functions: If $f$ is a real, $n$-variate affine function which is Boolean on the $n$-dimensional hypercube (that is, $f(x) \in \{ 0, 1 \}$ for $x \in \{ 0, 1 \}^n$), then $f(x) = 0$, $f(x) = 1$, $f(x) = x_i$ or $f(x) = 1 – x_i$. The same classification (essentially) holds if we restrict $\{ 0, 1\}^n$ to elements with Hamming weight $k$ if $n-k, k \geq 2$. If we replace $k$-sets of $\{ 1, \ldots, n \}$ by $k$-spaces in $V(n, q)$, the $n$-dimensional vector space over the field with $q$ elements, then suddenly even the simple question of classifying Boolean degree $1$ functions, here traditionally known as Cameron-Liebler classes, becomes seemingly hard to solve. We will discuss some results on low-degree Boolean functions in the vector space setting. Most notably, we will discuss how vector space Ramsey numbers, so extremal combinatorics, can be utilized in this finite geometrical setting.

In [1], Fabianski et. al. developed a simple, yet surprisingly powerful algorithmic framework to develop efficient parameterized graph algorithms. Notably they derive a simple parameterized algorithm for the dominating set problem on a variety of graph classes, including powers of nowhere dense classes and biclique-free classes. These results encompass a wide range of previously known results and often improve the best known running times. Similar results follow for the distance-r variation of dominating set and for independent set. The running time of the algorithm is closely tied to model-theoretic properties, i.e. stability and the Helly property. We build upon these results and develop a similar algorithm which only relies on the strong Helly property and does not need stability. For the dominating set problem, we get a parameterized algorithm that works (additionally to biclique-free classes and powers of nowhere dense classes) weakly gamma-closed classes while being simpler and faster than previously known results. In this talk, we introduce the basic framework, results by Fabianski et. al and connections to other areas. We discuss our new insights and possible research directions. [1] Grzegorz Fabianski, Michal Pilipczuk, Sebastian Siebertz, Szymon Torunczyk: Progressive Algorithms for Domination and Independence. STACS 2019