# A 3-regular matchstick graph of girth 5 consisting of 54 vertices

## TL;DR

This paper presents the construction of the smallest known 3-regular matchstick graph with girth 5, consisting of only 54 vertices, improving upon the previous smallest known example.

## Contribution

The authors construct and verify a new 54-vertex 3-regular matchstick graph with girth 5, reducing the known minimum size from 180 vertices.

## Key findings

- Constructed a 54-vertex graph with the desired properties
- Proved the geometric correctness of the constructed graph
- Reduced the known minimum size for such graphs from 180 to 54 vertices

## Abstract

In 2010 it was proved that a 3-regular matchstick graph of girth 5 must consist at least of 30 vertices. The smallest known example consisted of 180 vertices. In this article we construct an example consisting of 54 vertices and prove its geometrical correctness.

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Source: https://tomesphere.com/paper/1903.04304