Chordal Bipartite Graphs with High Boxicity

TL;DR

This paper disproves a conjecture by showing that some chordal bipartite graphs can have arbitrarily high boxicity, challenging previous assumptions about their geometric representations.

Contribution

The paper provides a counterexample family of chordal bipartite graphs with unbounded boxicity, refuting the conjecture that their boxicity is at most 2.

Findings

Chordal bipartite graphs can have arbitrarily high boxicity.

The conjecture that all chordal bipartite graphs have boxicity ≤ 2 is false.

Counterexamples form an infinite family of such graphs.

Abstract

The boxicity of a graph G is defined as the minimum integer k such that G is an intersection graph of axis-parallel k-dimensional boxes. Chordal bipartite graphs are bipartite graphs that do not contain an induced cycle of length greater than 4. It was conjectured by Otachi, Okamoto and Yamazaki that chordal bipartite graphs have boxicity at most 2. We disprove this conjecture by exhibiting an infinite family of chordal bipartite graphs that have unbounded boxicity.