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3-Regular Colored Graphs and Classification of Surfaces

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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 \(m\ge 1\). Using this, we present a simple proof of the classification of closed surfaces.

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Acknowledgements

The author would like to thank Rekha Santhanam for suggesting the problem. The author thanks Basudeb Datta for the proof of Lemma 3.2, and for many useful comments and suggestions which led to the current presentation of this article. The author also thanks Bhaskar Bagchi and anonymous referees for many helpful comments. The author is supported by NBHM, India for Postdoctoral Fellowship (Award No.: 2/40(49)/2015/R&D-II/11568).

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  1. Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore, 560 059, India

    Biplab Basak

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  1. Biplab Basak
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Correspondence to Biplab Basak.

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Basak, B. 3-Regular Colored Graphs and Classification of Surfaces. Discrete Comput Geom 58, 345–354 (2017). https://doi.org/10.1007/s00454-017-9861-3

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  • DOI: https://doi.org/10.1007/s00454-017-9861-3

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