On March 24, 2020, Kevin Hendrey from IBS discrete mathematics group presented a talk on his work on covering the Hamming permutation space by balls of small radius. The title of his talk was “Covering radius in the Hamming permutation space“.

## 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 presented his work on the minimum connectivity to force a minor isomorphic to a fixed forest in a large graph at the discrete math seminar

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”.

## 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.

## Welcome Kevin Hendrey, a new research fellow in the IBS discrete mathematics group

The IBS discrete mathematics group welcomes Dr. **Kevin Hendrey**, a new research fellow at the IBS discrete mathematics group from August 16, 2019.

He received his Ph.D. from the School of Mathematics, Monash University in Australia in 2019 under the supervision of Prof. David R. Wood. Welcome!