4-regular planar unit triangle graphs without additional triangles

TL;DR

This paper proves the existence of 4-regular planar graphs made solely of unit triangles, with the smallest such graph having at most 6422 triangles, advancing understanding of geometric graph constructions.

Contribution

It demonstrates the existence of 4-regular planar unit triangle graphs without extra triangles and provides an upper bound on their size.

Findings

Existence of 4-regular planar unit triangle graphs proven.

Smallest such graph has at most 6422 triangles.

No additional triangles are needed in these graphs.

Abstract

In this article we proof the existence of 4-regular planar unit-distance graphs consisting only of unit triangles without additional triangles. It is shown that the smallest number of unit triangles is $\leq$6422.