Perfect colorings of a design with 2-dimensional euclidean crystallographic symmetry group.

Abstract

This thesis introduces perfect colorings and the systematic way of generating perfect colorings. We study the perfect colorings for designs whose symmetry groups are 2-dimensional Euclidean crystallographic groups. Two-dimensional Euclidean crys- tallographic groups are discrete subgroups of the isometry group of the Euclidean plane. We classify crystallographic groups by their ranks. Crystallographic groups of rank 0, rank 1, and rank 2 are explained with details and examples. Theorems that related to the number of perfect colorings for designs with crystallographic symmetry groups are constructed and proved.