Minimal completely asymmetric (4; n)-regular matchstick graphs

TL;DR

This paper introduces the smallest known completely asymmetric (4;n)-regular matchstick graphs for 4≤n≤11, characterized by rigidity, asymmetry, and minimal vertices, expanding understanding of asymmetric geometric graph structures.

Contribution

It presents the first minimal completely asymmetric (4;n)-regular matchstick graphs for specified n values, detailing their construction and properties.

Findings

Identified minimal vertex counts for (4;n)-regular matchstick graphs

Demonstrated the existence of completely asymmetric structures

Provided explicit examples with detailed properties

Abstract

A matchstick graph is a graph drawn with straight edges in the plane such that the edges have unit length, and non-adjacent edges do not intersect. We call a matchstick graph $(m;n)$-regular if every vertex has only degree $m$ or $n$. In this article we present the latest known $(4;n)$-regular matchstick graphs for $4\leq n\leq11$ with a minimum number of vertices and a completely asymmetric structure. We call a matchstick graph completely asymmetric, if the following conditions are complied. 1) The graph is rigid. 2) The graph has no point, rotational or mirror symmetry. 3) The graph has an asymmetric outer shape. 4) The graph can not be decomposed into rigid subgraphs and rearrange to a similar graph which contradicts to any of the other conditions.