Bad News for Chordal Partitions

TL;DR

This paper proves that not all graphs can be partitioned into induced connected bipartite subgraphs with a chordal quotient, refuting a longstanding conjecture and its relaxations, impacting graph theory and Hadwiger's Conjecture.

Contribution

It demonstrates that the conjecture by Reed and Seymour regarding such partitions into bipartite subgraphs with a chordal quotient is false, even under relaxed conditions.

Findings

The conjecture is false for all graphs.

Relaxations of the conjecture also do not hold.

Implications for Hadwiger's Conjecture.

Abstract

Reed and Seymour [1998] asked whether every graph has a partition into induced connected non-empty bipartite subgraphs such that the quotient graph is chordal. If true, this would have significant ramifications for Hadwiger's Conjecture. We prove that the answer is `no'. In fact, we show that the answer is still `no' for several relaxations of the question.