Bad News for Chordal Partitions

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.