3-regular colored graphs and classification of surfaces

TL;DR

This paper studies 3-regular colored graphs motivated by crystallization theory, establishing uniqueness and classification results that lead to a simple proof of the classification of closed surfaces.

Contribution

It introduces an equivalence relation on 3-regular colored graphs and proves uniqueness and multiplicity results, providing a new simple proof of surface classification.

Findings

Unique contracted 3-regular colored graph for vertices 4m

Exactly two such graphs for vertices 4m+2

Simplified proof of closed surface classification

Abstract

Motivated by the theory of crystallizations, we consider an equivalence relation on the class of $3$-regular colored graphs and prove that up to this equivalence (a) there exists a unique contracted 3-regular colored graph if the number of vertices is $4m$ and (b) there are exactly two such graphs if the number of vertices is $4m+2$ for each $m\geq 1$. Using this, we present a simple proof of the classification of closed surfaces.