Strengthening strong immersions with Kempe chains

TL;DR

This paper introduces a novel approach using Kempe chains to strengthen strong immersions in properly colored graphs, potentially aiding in identifying complete graph minors related to Hadwiger's conjecture.

Contribution

It presents a new method to strengthen strong immersions via Kempe chains, offering a potential framework for analyzing Hadwiger's conjecture in k-chromatic graphs.

Findings

Kempe chains form edge-disjoint backbones in k-chromatic graphs.

Color permutations can organize these backbones into clique-like structures.

The approach suggests a template for identifying K_k minors in k-chromatic graphs.

Abstract

Every properly colored graph with $\chi(G)=k$ colors has edge-disjoint Kempe "backbones", Kempe chains anchored by color-critical vertices for each pair of colors. Certain color permutations arrange these backbones into a clique-like structure, a strengthening of strong immersions of complete graphs. This strengthened immersion is suggested as a template for identifying the disjoint subgraphs comprising Hadwiger's conjectured $K_k$ minor present in $k$-chromatic graphs.