Partial Chromatic Polynomials and Diagonally Distinct Sudoku Squares

TL;DR

This paper presents a simplified proof that the number of ways to complete partial Sudoku colorings is polynomial in the number of colors, and constructs Sudoku squares with unique diagonal entries for any size.

Contribution

It offers a new, simpler proof of a polynomial counting theorem for Sudoku graph colorings and constructs diagonally distinct Sudoku squares of arbitrary size.

Findings

Number of Sudoku completions is polynomial in the number of colors.

Constructed Sudoku squares with distinct diagonal entries for any size.

Provided a new proof technique for graph coloring enumeration.

Abstract

Sudoku grids can be thought of as graphs where the vertices are the squares of the grid, and edges join vertices in the same row, column, or sub-grid. A Sudoku puzzle corresponds to a partial proper coloring of the Sudoku graph. We provide a new and simpler proof of the theorem which states that the number of completions of partial colorings of a graph is a polynomial in the number of colors (originally due to Herzberg and Murty). Moreover, we construct Sudoku squares of arbitrary size with distinct entries on both diagonals (a similar proof was first published by Keedwell, unknown to the author).