Combinatorial Properties of Triangle-Free Rectangle Arrangements and the Squarability Problem

TL;DR

This paper explores the combinatorial and geometric properties of triangle-free rectangle arrangements, establishing their correspondence with planar graphs and investigating conditions under which these arrangements can be transformed into square arrangements.

Contribution

It introduces a bijection between triangle-free rectangle arrangements and 4-orientations of planar multigraphs, and examines squarability of such arrangements under certain conditions.

Findings

Triangle-free rectangle arrangements correspond to 4-orientations of planar multigraphs.

Every triangle-free planar graph can be realized as a contact graph of such arrangements.

Rectangle arrangements pierced by a horizontal line can be squarable under specific conditions.

Abstract

We consider arrangements of axis-aligned rectangles in the plane. A geometric arrangement specifies the coordinates of all rectangles, while a combinatorial arrangement specifies only the respective intersection type in which each pair of rectangles intersects. First, we investigate combinatorial contact arrangements, i.e., arrangements of interior-disjoint rectangles, with a triangle-free intersection graph. We show that such rectangle arrangements are in bijection with the 4-orientations of an underlying planar multigraph and prove that there is a corresponding geometric rectangle contact arrangement. Moreover, we prove that every triangle-free planar graph is the contact graph of such an arrangement. Secondly, we introduce the question whether a given rectangle arrangement has a combinatorially equivalent square arrangement. In addition to some necessary conditions and counterexamples, we show that rectangle arrangements pierced by a horizontal line are squarable under certain sufficient conditions.