On the existence of 4-regular matchstick graphs

TL;DR

The paper proves that there exist 4-regular matchstick graphs for every $n\ge 63$, resolving a longstanding gap by building graphs from $(2;4)$-regular blocks with two degree-2 vertices and a fixed set of rigid/flexible components. The key technique is a vertex-sum construction: for a graph built from $k$ blocks, $|V| = \sum|V_i| - k$ and $|E| = 2|V|$, enabling controlled assembly of 4-regular graphs across ranges $63 \le n \le 120$ (with a few exceptions) and then extending to infinite families for $n>93$ by inserting flexible subgraphs. The work combines exhaustive block-based constructions, rigidity analysis verified by the Matchstick Graphs Calculator (MGC), and cataloging of small-$n$ examples to establish existence and provide a practical toolkit for generating numerous distinct graphs. This advances both the theoretical understanding and practical cataloging of unit-distance planar graphs with fixed degree sequences, with potential implications for geometric graph theory and related applications.

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 $n \geq 63$.

Figures (22)

• Figure 1: 52 vertices
• Figure 2: 54 vertices
• Figure 3: 57 vertices
• Figure 4: 60 vertices
• Figure 6: 22 vertices

Theorems & Definitions (20)

• Theorem 1
• Definition 1
• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Remark 1
• Theorem 2
• proof
• Theorem 3