Events

Title: Graph Embeddings and Map Colorings

Abstract: The famous  Map Color Theorem says that the chromatic number of a surface of Euler characteristic $c<0$ is equal to $\displaystyle \left\lfloor \frac{1}{2}\left(7+\sqrt{49-24c}\right)\right\rfloor $. This was proved in 1969 by Ringel and Youngs who showed that $K_n$ can be embedded on surfaces of Euler characteristic $c$ such that $\displaystyle n= \left\lfloor \frac{1}{2}\left(7+\sqrt{49-24c}\right)\right\rfloor $. This leads to the study about the  genus distribution of a graph $G$, that is, the number of embeddings of $G$ on surfaces. This talk will go through some recent results about genus distributions of bouquets and cubic graphs.  Some results and conjectures will also be given about the distribution of the  chromatic number of a random map on a given surface.

Title: On the use of senders in Ramsey Theory

Abstract: In this talk I will introduce and investigate some parameters in Graph Ramsey theory, beyond the traditional Ramsey numbers. A crucial ingredient for their analysis is the existence of gadget graphs, called signal senders, that were initially developed by Burr, Erdős and Lovász in 1976. I will explain their origin, properties, and try to convey their surprising strength. Using probabilistic methods, we will see how to build such gadgets, and how to use them to prove some theorems, previously out of reach without these tools.

Title: A relaxation of Woodall’s conjecture

Abstract: In a directed graph, a directed cut (dicut for short) is a cut where all arcs are directed from one shore to the other; a directed join (dijoin for short) is a set of arcs whose contraction makes the digraph strongly connected. The celebrated Lucchesi–Younger theorem states that for any directed graph the size of the smallest dijoin equals the maximum number of pairwise disjoint dicuts. Woodall’s conjecture posits that the size of the smallest dicut equals the maximum number of pairwise disjoint dijoins. 

Title: Bipartite Perfect Matching is in Quasi-NC, Part II

Abstract: Mulmuley, Vazirani, and Vazirani gave a randomized parallel algorithm for checking whether a perfect matching exists in a graph. In doing so, they came up with the infamous isolation lemma, which found several uses in other areas of computer science. The isolation lemma is inherently randomized, and it has been a long-standing open problem to derandomize the lemma. In this talk, I will go over the breakthrough work of Fenner, Gurjar, and Thierauf where they almost completely derandomize the isolation lemma in the special case when applied to the bipartite perfect matching problem. In doing so, they give a deterministic parallel algorithm for perfect matching that uses a quasi-polynomial number of processors.

Title: Supersingular isogeny graphs, modular curves and Galois Representations.

Abstract: In this talk we will discuss the remarkable interconnection among supersingular elliptic curves, modular curves and Galois representations with a focus on cryptographic applications.

Fulkerson 100 is a workshop organized by the Dept. of Combinatorics & Optimization (C&O) from July 17-19, 2024 at the University of Waterloo, to celebrate Fulkerson's legacy and impact in discrete mathematics, especially in the fields of graph theory, optimization, and operations research. Fulkerson 100 will feature invited talks in graph theory, combinatorics, optimization, and theoretical computer science, given by some of the foremost researchers in these areas, as well as lightning talks and a poster session devoted to students and postdocs. By bringing together various leading researchers in discrete mathematics with junior researchers and students, the workshop aims to boost research in the areas pioneered by Fulkerson, while commemorating his vision and contributions.