Graph Coloring Approach to Solve Sudoku Problems

Abstract

The Classic Sudoku Puzzle's intrinsic combinatorial difficulty has captivated mathematicians and puzzle enthusiasts for decades. We investigate a graph coloring method for resolving different sized Sudoku puzzles, specifically concentrating on the $4\times 4$ and $9\times 9$ grid sizes. Through the methodical application of graph coloring techniques, our goal is to analyze the complexities of these puzzles by providing a detailed and sequential guide that simplifies the process of solving these puzzles. We also examine the implications of the chromatic number equivalence on the order of the grid, puzzle complexity, pattern formation, and solution strategies, seeking to unravel the underlying mathematical structures governing these specialized Sudoku instances. Additionally, we extend our attention to the problem formulation step, where we can compute the vertices and edges of different-sized Sudoku puzzles. Furthermore, our discoveries indicate that the strategic application of this approach holds potential benefits for Sudoku puzzles irrespective of their size. This insight illuminates the intricate and elegant nature of Sudoku, portraying it not only as a mathematical challenge but also as a computational one.