Abstract :

Crystallization is a graph-theoretical tool to study topological and combinatorial properties of PL manifolds. Motivated by the theory of crystallizations, we consider an equivalence relation on the class of 3-regular colored graphs. In this talk, we shall 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 $mgeq 1$. Using this, we shall present a simple proof of the classification of closed surfaces. At the end of this talk, we shall discuss some important properties and results of crystallization in higher dimension. Then, we shall quickly go through the higher dimensional analog of genus, namely regular genus.