Erdős and Pósa proved in 1965 that there is a duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. Such a duality does not hold if we restrict to odd cycles. However, in 1999, Reed proved an analogue for odd cycles by relaxing packing to half-integral packing. We prove a far-reaching generalisation of the theorem of Reed; if the edges of a graph are labelled by finitely many abelian groups, then there is a duality between the maximum size of a half-integral packing of cycles whose values avoid a fixed finite set for each abelian group and the minimum size of a vertex set hitting all such cycles.

A multitude of natural properties of cycles can be encoded in this setting, for example cycles of length at least $\ell$, cycles of length $p$ modulo $q$, cycles intersecting a prescribed set of vertices at least $t$ times, and cycles contained in given $\mathbb{Z}_2$-homology classes in a graph embedded on a fixed surface. Our main result allows us to prove a duality theorem for cycles satisfying a fixed set of finitely many such properties.

This is joint work with J. Pascal Gollin, Ken-ichi Kawarabayashi, O-joung Kwon, and Sang-il Oum.

Our problem can be described in terms of a two player game, played with the set $\mathcal{S}_n$ of permutations on $\{1,2,\dots,n\}$. First, Player 1 selects a subset $S$ of $\mathcal{S}_n$ and shows it to Player 2. Next, Player 2 selects a permutation $p$ from $\mathcal{S}_n$ as different as possible from the permutations in $S$, and shows it to Player 1. Finally, Player 1 selects a permutation $q$ from $S$, and they compare $p$ and $q$. The aim of Player 1 is to ensure that $p$ and $q$ differ in few positions, while keeping the size of $S$ small. The function $f(n,s)$ can be defined as the minimum size of a set $S\subseteq \mathcal{S}_n$ that Player 1 can select in order to gaurantee that $p$ and $q$ will differ in at most $s$ positions.

I will present some recent results on the function $f(n,s)$. We are particularly interested in determining the value $f(n,2)$, which would resolve a conjecture of Kézdy and Snevily that implies several famous conjectures for Latin squares. Here we improve the best known lower bound, showing that $f(n,2)\geqslant 3n/4$. This talk is based on joint work with Ian M. Wanless.

On September 10, 2019, Kevin Hendrey at IBS Discrete Mathematics Group presented his work on the minimum connectivity to force a minor isomorphic to a fixed forest in a large graph. The title of his talk was “The minimum connectivity forcing forest minors in large graphs”.

Given a graph $G$, we define $\textrm{ex}_c(G)$ to be the minimum value of $t$ for which there exists a constant $N(t,G)$ such that every $t$-connected graph with at least $N(t,G)$ vertices contains $G$ as a minor. The value of $\textrm{ex}_c(G)$ is known to be tied to the vertex cover number $\tau(G)$, and in fact $\tau(G)\leq \textrm{ex}_c(G)\leq \frac{31}{2}(\tau(G)+1)$. We give the precise value of $\textrm{ex}_c(G)$ when $G$ is a forest. In particular we find that $\textrm{ex}_c(G)\leq \tau(G)+2$ in this setting, which is tight for infinitely many forests.