Events

Title: Decomposing graphs and hypergraphs into complete bipartite subgraphs

Abstract:

The problem of decomposing (partitioning or covering) graphs into complete bipartite subgraphs (bicliques) has a long history. In this talk, I will describe the basic results including the use of spectral methods, the extension of the problem to hypergraphs and present some of the open problems in this area.

Title: Reduced bandwidth: a qualitative strengthening of twin-width in minor-closed classes (and beyond)

Abstract:

In a reduction sequence of a graph, vertices are successively identified until the graph has one vertex. At each step, when identifying $u$ and $v$, each edge incident to exactly one of $u$ and $v$ is coloured red.

Title: Sorting probabilities for Young diagrams and beyond

Abstract:

Sorting probability for a partially ordered set P is defined as the min |Pr[x<y] - Pr[y<x]| going over all pairs of elements x,y in P, where Pr[x<y] is the probability that in a uniformly random linear extension (extension to total order) x appears before y.

The celebrated 1/3-2/3 conjecture states that for every poset the sorting probability is at most 1/3, i.e. there are two elements x and y, such that 1/3\leq Pr[x<y] \leq 2/3.

Title: The chromatic number of triangle-free hypergraphs

Abstract:

A classical result of Johansson showed that for any triangle-free graph $G$ with maximum degree $\Delta$, it chromatic number is $O(\Delta/\log\Delta)$. This result was later generalized to all rank 3 hypergraphs due to the work of Copper and Mubayi. In this talk, I will present a common generalization of the above results to all hypergraphs, and this is sharp apart from the constant.

Title: Oriented Cayley Graphs with all eigenvalues being integer multiples of $\sqrt{\Delta}$

Abstract:

Let $G$ be a finite abelian group. An oriented Cayley graph on $G$ is a Cayley digraph $X(G,C)$ such that $C \cap (-C)=\emptyset$. Consider the $(0,1,-1)$ skew-symmetric adjacency matrix of an oriented Cayley graph $X=X(G,C)$.

Title: Minimal induced subgraphs of two classes of 2-connected non-Hamiltonian graphs

Abstract:

Finding sufficient conditions for a class of graphs to be Hamiltonian is an old problem, with a wide variety of conditions such as Dirac's degree condition and Whitney's theorem on 4-connected planar triangulations. We discuss some past results on sufficient conditions for Hamiltonicity involving the exclusion of fixed induced subgraphs, and some properties of the graphs involved in such results.

Title: Multiplying quantum Schubert polynomials using combinatorics

Abstract:

Schubert polynomials are a very interesting family of polynomials in algebraic geometry due to their relation with the cohomology of the flag variety. Moreover, they are also very interesting from a combinatorial point of view because they can be considered generalizations of Schur functions. 

Title: SAT Solving with Computer Algebra for Combinatorics

Abstract:

This talk will describe a method of solving combinatorial problems by coupling Boolean satisfiability (SAT) solvers with computer algebra systems (CASs), thereby combining the search and learning power of SAT solvers with the expressiveness and mathematical sophistication of CASs. 

Title: Yet Another Proof of the Erdős-Ko-Rado Theorem

Abstract:

We give a short new algebraic proof of the Erdős-Ko-Rado theorem, that for k < n/2, the largest families of k-sets of an n-element set such that any two of its members intersect are precisely those families composed of all k-sets that contain some fixed element.

Title: A local version of Hadwiger’s Conjecture

Abstract:

In 1943, Hadwiger famously conjectured that graphs with no $K_t$ minors are $t-1$ colourable. There has also been significant interest in several variants of the problem, such as list colouring or only forbidding certain classes of minors.