Sudoku Scheming Am Algebraic Combinatorial Approach to Discovering Properties of Sudoku Graphs using Association Schemes

Ziva Myer

Date of Award

2011

Document Type

Thesis

Degree Name

Bachelors

Department

Natural Sciences

First Advisor

Piomenidou, Eirini

Keywords

Sudoku, Graph Theory, Association Schemes

Area of Concentration

Mathematics

Abstract

In a 2009 paper, Dahl introduced S-permutation matrices and the operation S-interchange performed on these matrices. These concepts combine to make the Sudoku Graph of order N (which we call ?N). Dahl offered an upperbound and lowerbound for the diameter of the Sudoku Graph. Using his work as a starting point, we focused mainly on the Sudoku Graphs for N=4,9. We found ?4 to be the hypercube graph, and studied the association scheme, the Hamming Scheme, on its distances, showing it to be distance regular. Forming an alternative notation, Block Coordinate Notation (BCN), for S-permutation matrices, we wrote a program in Python to assist with counting distances in the much larger graph ?9. We found that is not distance regular, even though the distances are consistent for any starting vertex. Lastly, we proved that the diameter of ?9 is 12 and proposed a general conjecture for the diameter of ?N.

Rights

This bibliographic record is available under the Creative Commons CC0 public domain dedication. The New College of Florida, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.

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