Completion of the Partial Latin Squares and Coloring, and a Note on the Trades in Steiner Triple Systems

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Soltani, Farhad | 2018

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 51233 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Mahmoodian, Abadollah; Daneshgar, Amir
Abstract:
The problem of completing partial latin squares to latin squares of the same order has been studied for many years. Any latin square of order n is equivalent to an n-coloring of the graph Kn□Kn and any partial latin square of order n is equivalent to a partial coloring of the graph Kn□Kn. In this thesis at first, we will focus on the study of the unique completion of partial coloring of induced subgraphs of the graph Km□Kn.The final section of this thesis is a study on Steiner triple systems. A graph G is formed with the Steiner triple systems of a given order on a given base set as the vertices, and vertices joined by an edge when the corresponding Steiner triple systems can be converted to each other by applying a Pasch trade(minimal trade). Unfortunately this graph is not connected. By generalizing the consept of Steiner triple systems (to Steiner triple system with at most one negative bloch), it is conjectured that this generalized Steiner triple systems (of same order) can be converted to each other by applying some finite sequence of Pasch trades [4, 8]
Keywords:
Partial Latin Square ; Steiner Triple System (STS) ; Latin Trade ; Latin Square Method ; Partial Latin Square Completion