Interval representation of balanced separators in graphs avoiding a minor

We show that for any sufficiently large graph $G$ avoiding $K_k$ as a minor, we can map vertices $v\in V(G)$ to intervals $I(v)\subseteq [0,1]$ so that

1. $I(u)\cap I(v)\ne \mathrm{\varnothing }$ for each edge $uv$,
2. the sum of the squares of the lengths of these intervals is $\mathcal{O}(k^6\log k)$, and
3. the average distance between the intervals is at least $\frac{1}{25}$

Balanced separators of $G$ of sublinear size (with various additional properties) can be read off this representation.

This is joint work with Zdeněk Dvořák and Jakub Pekárek

Recording