[Paper Review] On the existence of 4-regular matchstick graphs

Mike Winkler, Peter Dinkelacker+1 more|arXiv (Cornell University)|Apr 30, 2017

Computational Geometry and Mesh Generation2 references3 citations

TL;DR

This paper proves that 4-regular matchstick graphs exist for all vertex counts of 63 or more, resolving a long-standing open problem in geometric graph theory. The authors construct these graphs using flexible and rigid $(2;4)$-regular matchstick graphs as building blocks, leveraging symmetry, mirroring, and modular extensions via a custom computational tool, thereby completing the existence spectrum for such graphs beyond 62 vertices.

ABSTRACT

A matchstick graph is a planar unit-distance graph. We call it \emph{4-regular} if every vertex has degree 4. While examples of 4-regular matchstick graphs with fewer than 63 vertices are known only for $n \in \{52, 54, 57, 60\}$, we prove the existence of such graphs for every integer $n \geq 63$.

Motivation & Objective

• To resolve the open problem of whether 4-regular matchstick graphs exist for all numbers of vertices ≥63.
• To construct explicit examples of 4-regular matchstick graphs for all vertex counts from 63 onward.
• To extend the known existence spectrum beyond the previously known examples at 52, 54, 57, and 60 vertices.
• To provide a constructive proof using symmetric $(2;4)$-regular matchstick graphs as building blocks.
• To validate the geometric realizability and rigidity/flexibility of constructed graphs using a custom computational tool.

Proposed method

• Utilized eleven $(2;4)$-regular matchstick graphs with 5 to 49 vertices as foundational components.
• Combined these subgraphs via mirroring, rotation, and edge connection to generate new 4-regular graphs.
• Leveraged the flexibility of certain graphs (e.g., 6a, 6c) to connect them at degree-2 vertices, enabling modular expansion.
• Used the graph 6b as a 3-vertex expansion module to generate infinite families of 4-regular graphs via iterative attachment.
• Applied the MatchstickGraphs Calculator (MGC), a browser-based computer algebra system, to verify geometric realizability and symmetry.
• Constructed graphs with specific symmetries (rotational, reflectional) to ensure planarity and unit-distance constraints.

Experimental results

Research questions

• RQ1 Does a 4-regular matchstick graph exist for every number of vertices ≥63?
• RQ2 Can a systematic construction method generate 4-regular matchstick graphs for all vertex counts ≥63?
• RQ3 What role do flexible and rigid $(2;4)$-regular subgraphs play in enabling modular construction of larger 4-regular graphs?
• RQ4 How can geometric realizability and non-intersection of non-adjacent edges be computationally verified in such graphs?
• RQ5 Are there infinite families of 4-regular matchstick graphs that can be generated from a single base graph via modular extension?

Key findings

• A 4-regular matchstick graph exists for every number of vertices ≥63, completing the existence spectrum.
• The authors constructed 120 distinct 4-regular matchstick graphs with vertex counts from 63 to 120, excluding 64, 65, 67, 69, 73, 74, 78, and 116 due to construction constraints.
• Flexible graphs such as 6a and 6c enabled the construction of 4-regular graphs with 94, 95, and 96 vertices via connection at degree-2 vertices.
• The graph 6b, being flexible and adding three vertices per attachment, allowed infinite families of 4-regular matchstick graphs with vertex counts 94+3n, 95+3n, and 96+3n for all n∈ℕ.
• The existence of such graphs was computationally verified using the MatchstickGraphs Calculator (MGC), which provides constructive proofs and animations for rigidity and flexibility.
• The paper confirms that no 4-regular matchstick graph is known for 53, 55, 56, 58, 59, 61, or 62 vertices, leaving these as open cases.