Graph minimization, focusing on the example of 5-chromatic unit-distance graphs in the plane

TL;DR

This paper presents a new graph minimization technique that preserves specific properties, demonstrated by reducing 5-chromatic unit-distance graphs in the plane to a smaller form with 509 vertices.

Contribution

The paper introduces a novel graph minimization method with an effective property-preservation check, applied to 5-chromatic unit-distance graphs.

Findings

Reduced 5-chromatic unit-distance graphs to 509 vertices

Developed an effective procedure for property checking in graph minimization

Achieved a smaller, optimized graph with 2442 edges

Abstract

We introduce a new graph minimization method, in which it is required to preserve some graph property and there is an effective procedure for checking this property. We applied this method to minimize 5-chromatic unit-distance graphs and obtained a graph with 509 vertices and 2442 edges.