Kevin Hendrey, A unified half-integral Erdős-Pósa theorem for cycles in graphs labelled by multiple abelian groups

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.

Kevin Hendrey, Covering radius in the Hamming permutation space

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.

Kevin Hendrey, 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.