@article{KM21kempechainsrootedminors,
title = {Kempe chains and rooted minors},
author = {Kriesell, Matthias and Mohr, Samuel},
note = {submitted.},
archivePrefix = {arXiv},
eprint = {1911.09998},
biburl = {http://samuel-mohr.de/files/bib/8.bib},
abstract = { A (minimal) {\em transversal} of a partition is a set which contains exactly one element from each member of the partition and nothing else. A \emph{coloring} 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 coloring {$\mathcal{C}$} of some graph {$G$},
is there a partition {$\mathcal{H}$} of a subset of {$V(G)$} into connected sets such that {$T$} is a transversal of {$\mathcal{H}$}
and such that two sets of {$\mathcal{H}$} are adjacent if their corresponding vertices from {$T$} are connected by a path in {$G$} using only two colors?\\
It has been conjectured by the first author that for any transversal {$T$} of a coloring {$\mathcal{C}$} of order {$k$} of some graph {$G$} such that any
pair of color classes induces a connected graph, there exists such a partition {$\mathcal{H}$} with pairwise adjacent sets (which would prove Hadwiger's conjecture for the class of uniquely optimally colorable graphs);
this is open for each {$k \geq 5$}, here we give a proof for the case that {$k=5$} and the subgraph induced by {$T$} is connected.
Moreover, we show that for {$k\geq 7$}, it is not sufficient for the existence of {$\mathcal{H}$} as above just to force any two transversal vertices to be connected by a 2-colored path. }
}