Spectrum of free-form Sudoku graphs

TL;DR

This paper investigates the spectral properties of free-form Sudoku graphs, focusing on eigenvalues, integrality, and how these properties evolve under grid scaling operations.

Contribution

It introduces a spectral analysis framework for free-form Sudoku graphs and examines how eigenvalues change with grid scaling, a novel approach in puzzle graph theory.

Findings

Eigenvalues of free-form Sudoku graphs are studied, with a focus on conditions for integrality.

Eigenvalues and eigenspaces evolve predictably under grid 'blow up' operations.

The spectral properties provide insights into the structure and symmetries of Sudoku puzzles.

Abstract

A free-form Sudoku puzzle is a square arrangement of m times m cells such that the cells are partitioned into m subsets (called blocks) of equal cardinality. The goal of the puzzle is to place integers 1,...,m in the cells such that the numbers in every row, column and block are distinct. Represent each cell by a vertex and add edges between two vertices exactly when the corresponding cells, according to the rules, must contain different numbers. This yields the associated free-form Sudoku graph. This article studies the eigenvalues of free-form Sudoku graphs, most notably integrality. Further, we analyze the evolution of eigenvalues and eigenspaces of such graphs when the associated puzzle is subjected to a "blow up" operation, scaling the cell grid including its block partition.