Colorings of Latin Square Graphs and Block Designs

Please enable javascript in your browser.

Mahmoodian, Ebadollah | 2009

1326 Viewed

Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 40107 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Mortezaeefar, Masood
Abstract:
In the first part of this thesis we study Latin Square Graphs defined as follows: to each cell of a latin square L of order n we associate a vertex of the graph (which we call L3(n)) with two vertices u and v adjacent if and only if their respective cells are in the same row or column or contain the same symbol. We investigate the coloring problem associated with these graphs and explore the relationship between this problem and some other combinatorial objects such as strongly regular graphs and transversals and obtain some upper bounds. We use a computer search to find the chromatic number of all latin squares of order less than 8. We determine the chromatic number of some special classes of latin squares such as cyclic latin squares. In the second part we concentrate on the problem of coloring Block Designs and report the results of several numerical experiments. For every STS of order not exceeding 15, we obtain its coloring patterns, and we analyse this data to show some interesting results regarding the distribution of the patterns. We also compute the minimum defining set for each pattern.
Keywords:
Block Design ; Defining Set ; Latin Square Method ; Defining Set ; Graph Coloring ; Latin Square Graph ; Steiner Triple System (STS) ; Color Classes

No TOC