Representing a cubic graph as the intersection graph of axis-parallel boxes in three dimensions

TL;DR

This paper demonstrates that all graphs with maximum degree 3 can be represented as intersection graphs of 3D axis-parallel boxes, with a linear-time construction ensuring boundary contact for intersecting boxes.

Contribution

It provides a linear-time method to represent maximum degree 3 graphs as intersection graphs of 3D boxes with boundary contact.

Findings

All maximum degree 3 graphs can be represented as 3D box intersection graphs.

The construction runs in linear time.

Intersecting boxes only touch at boundaries.

Abstract

We show that every graph of maximum degree 3 can be represented as the intersection graph of axis parallel boxes in three dimensions, that is, every vertex can be mapped to an axis parallel box such that two boxes intersect if and only if their corresponding vertices are adjacent. In fact, we construct a representation in which any two intersecting boxes just touch at their boundaries. Further, this construction can be realized in linear time.