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On the Latin Square of Groups and their Coloring

Shokri, Kianoosh | 2015

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 47933 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Mahmoodian, Ebadollah
  7. Abstract:
  8. A Latin square of order n, is an n n array filled with n different symbols, each occurring exactly once in each row and exactly once in each column.Two cells of a Latin square are independent when they are not in the same row, or in the same column and they are not the same symbol. If they are not independent, they are called dependent. A k-coloring of a Latin square is assigning k colors to its cells where no two dependent cells have the same color. The smallest k for which we have a k-coloring for a Latin square L with k colors is called chromatic number of L and we denote it by L). If we consider the Cayley table of an arbitrary finite group of order n, then we have a Latin square of order n. Our goal is to investigate the chromatic number of Latin squares coming from groups. A transversal of a Latin square L, is a set of n cells of L, where no two of these cells are in the same row or in the same column or they have the same symbols. Let (G;) be an arbitrary finite group and be a permutation of elements of G. is called a complete mapping if: G ! G, where (x) = x (x) is a permutation of elements of G. Each transversal of a Latin square of a group is equal to a complete mapping of that group. Any Latin square of a group of odd order has a transversal. Having ransversal or complete mapping in a Latin square of arbitrary group is depended to the 2-Sylow subgroup of that group. If an arbitrary finite group has noncyclic or trivial 2-Sylow subgroup, then it has a complete mapping. In this thesis, the existence of a complete mapping in abelian and nonabelian groups of order less than 30 and also some special classes of groups is investigated. All groups with complete mapping have chromatic number equal to their order. For groups of order n which have no complete mapping, the chromatic number can not be n+1. We present some results about chromatic number of groups without complete mapping
  9. Keywords:
  10. Transversal ; Latin Square Method ; Groups Latin Square ; Complete Mapping ; Latin Square Chromatic Number

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