3-regular colored graphs and classification of surfaces
Biplab Basak

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 -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 and (b) there are exactly two such graphs if the number of vertices is for each . Using this, we present a simple proof of the classification of closed surfaces.
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