Let \(H\) be a planar graph. We prove that there exists a constant \(c:=c(H)\), such that for all \(k \in \mathbb N\) and all graphs \(G\), either \(G\) contains \(k\) vertex-disjoint subgraphs each containing \(H\) as a minor, or there is a subset \(X\) of at most \(c k \log k\) vertices such that \(G-X\) has no \(H\) minor.
This bound is best possible, up to the value of \(c\),
and improves upon the bounds of Robertson and Seymour, and of Chekuri and Chuzhoy. Joint work with Tony Huynh, Gwenaël Joret and Jean-Florent Raymond.
Seymour's conjecture on 2-connected graphs of large pathwidth ±
In the first paper of their graph minors series, Robertson and Seymour
proved that for every forest \(F\), there exists a constant \(p\) such that
every graph with pathwidth at least \(p\) contains \(F\) as a minor. Since
forests have unbounded pathwidth, this implies that a minor-closed
class of graphs has unbounded pathwidth if and only if it includes all
forests. However, these certificates of large pathwidth are not
2-connected, so it is natural to ask for which minor-closed classes \(C\),
does every 2-connected graph in \(C\) have bounded pathwidth?
In 1993, Paul Seymour proposed the following answer. A graph \(H\) is an
apex-forest if \(H-v\) is a forest for some vertex \(v\) of \(H\). A graph is
outerplanar if it has an embedding in the plane with all the vertices
on the outerface. These two graph classes are relevant since they both
contain 2-connected graphs with arbitrarily large pathwidth. Seymour
conjectured the following converse holds: For every apex-forest \(H_1\)
and outerplanar graph \(H_2\) there is an integer \(p\) such that every
2-connected graph of pathwidth at least \(p\) contains \(H_1\) or \(H_2\) as a
In this talk I will outline a proof of this conjecture obtained
jointly with Tony Huynh, Piotr Micek, and David R. Wood. An
independent proof was recently obtained by Dang and Thomas.
We say that a graph property is expressible in the "first order
language of graphs" if it can be written as a logic sentence using the
universal and existential quantifiers with variables ranging over the
nodes of the graph, the usual connectives AND, OR, NOT, parentheses
and the relations = and ~, where x ~ y means that x and y share an
For example, the property that G contains a triangle can be written as
Exists x,y,z : (x ~ y) AND (x ~ z) AND (y ~ z).
A classical result of Glebskii et al. 1969 and indepently Fagin
1976 states that if one samples a graph on n vertices uniformly at
random from all such graphs then every first order expressible graph
property holds with probability tending to either zero or one (as n
tends to infinity).
Since then, first order expressible properties have been studied
extensively on the most commonly studied model of random graphs, the
A number of very attractive and surprising results have been obtained,
and by now
we have a fairly full description of the behaviour of first order expressible
properties on this model.
After describing some of these results, time permitting, I will
discuss some recent and ongoing work where we try to see what happens
both in other models of random graphs (such as random planar graphs,
random perfect graphs, random geometric graphs, ...) and when one
considers other (i.e. "higher order") logics.
(Based on joint works with S. Haber, P. Heinig, M. Noy, A. Taraz)
The Erdős-Hajnal Conjecture states that for every graph \(F\) there is a constant \(c>0\) such that every graph on \(n\) vertices with no induced copy of \(F\) contains a clique or stable set of size at least \(n^c\). The conjecture is known to hold for only a few graphs \(F\), but there has been recent progress in showing that it holds when both a graph \(F\) and its complement are excluded. We prove a conjecture of Liebenau, Pilipczuk, Seymour and Spirkl, that this holds whenever \(F\) is a tree. Joint work with Maria Chudnovsky, Paul Seymour and Sophie Spirkl.
I will present an algebraic approach introduced by Jensen and Thomassen
to study the number of colourings of a graph, more specifically to obtain
sufficient conditions for a graph to have an exponential number of them (in terms
of the number of vertices).
Several conjectures will be considered, and we'll focus particularly on a conjecture
of Thomassen that every triangle-free planar graph has an exponential number of 3-colourings.
The talk is based on a joint work with S. Norin and a joint work with Z. Dvořák.