Kempe Chains and Rooted Minors

Abstract

A (minimal) transversal of a partition is a set which contains exactly one element from each member of the partition and nothing else. A 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 $\mathfrak{C}$ 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 such that two sets of $\mathfrak{H}$ are adjacent if their corresponding vertices from $T$ are connected by a path in $G$ using only two colors? It has been suggested by the first author to study the following question: for any transversal $T$ of a coloring $\mathfrak{C}$ of order $k$ of some graph $G$ such that any pair of color classes induces a connected graph, does there exist such a partition $\mathfrak{H}$ with pairwise adjacent sets (which would prove Hadwiger's Conjecture for the class of uniquely optimally colorable graphs)? This is open for small $k \geq 5$, 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 $\mathfrak{H}$ as above just to force any two transversal vertices to be connected by a 2-colored path.