The Plane-Width of Graphs

TL;DR

This paper explores the concept of plane-width in graphs, establishing its relationship with chromatic numbers and other geometric and combinatorial properties, and analyzing its behavior under various graph operations.

Contribution

It introduces new connections between plane-width, chromatic number, circular chromatic number, and packing problems, and studies its behavior under graph operations.

Findings

Plane-width relates to chromatic and circular chromatic numbers.

Connections established with packing unit discs and other geometric problems.

Behavior of plane-width under graph operations analyzed.

Abstract

Map vertices of a graph to (not necessarily distinct) points of the plane so that two adjacent vertices are mapped at least a unit distance apart. The plane-width of a graph is the minimum diameter of the image of the vertex set over all such mappings. We establish a relation between the plane-width of a graph and its chromatic number, and connect it to other well-known areas, including the circular chromatic number and the problem of packing unit discs in the plane. We also investigate how plane-width behaves under various operations, such as homomorphism, disjoint union, complement, and the Cartesian product.