@article{M21rootedstructures,
title = {Rooted Structures in Graphs},
author = {Mohr, Samuel},
journal = {Proceedings of the 3rd BYMAT Conference},
% number = {3},
pages = {95--98},
volume = {Vol.~2},
year = {2021},
url = {https://temat.es/monograficos/issue/view/vol-2},
biburl = {http://samuel-mohr.de/files/bib/extabstr5.bib},
abstract = {A \emph{transversal} of a partition is a set which contains exactly one element from each member of the partition and nothing else.
A \emph{colouring} of a graph is a partition of its vertex set into anticliques, that is, sets of pairwise nonadjacent vertices.
We study the following problem: Given a transversal $T$ of a proper colouring {$\mathcal{C}$} of some graph {$G$},
is there a partition {$\mathfrak{H}$} of a subset of {$V(G)$} into connected sets such that {$T$} is a transversal of {$\mathfrak{H}$}
and any two distinct sets of {$\mathfrak{H}$} are adjacent?
It has been conjectured by Matthias Kriesell~\href{https://doi.org/10.1002/jgt.22056}{[Journal of Graph Theory 85.1 (2017), pp. 207--216]} that for any transversal {$T$} of a colouring {$\mathcal{C}$} of order {$k$} of some graph {$G$} such that any
pair of colour classes induces a connected subgraph, there exists such a partition {$\mathfrak{H}$} with pairwise adjacent sets.
This would prove Hadwiger's conjecture for the class of uniquely optimally colourable graphs;
however it is open for each {$k \geq 5$}.
This paper will provide an overview about the stated conjecture.
It extracts associated results from my \href{https://www.db-thueringen.de/receive/dbt_mods_00045876}{PhD thesis} and the related papers, summarises their relevance to the stated problem, and discusses some unsuccessful attempts.
}
}