Kempe Chains and Rooted Minors

TL;DR

This paper investigates a graph coloring problem related to Kempe chains and rooted minors, proving a specific case for five colors and showing limitations for larger numbers of colors.

Contribution

It proves the existence of a certain partition for five colors when the induced subgraph is connected, and demonstrates that for seven or more colors, connectivity alone is insufficient.

Findings

Proved the case for five colors with connected induced subgraph.

Showed that for seven or more colors, connectivity does not guarantee the partition.

Extended understanding of Kempe chains and rooted minors in graph coloring.

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.