Extremal and monotone behaviour of the Sudoku number and related critical set parameters

TL;DR

This paper explores the extremal and monotone properties of the Sudoku number and related critical set parameters, providing characterizations, answering open questions, and extending results to hypergraph colouring.

Contribution

It characterizes graphs attaining extremal values of four Sudoku-related parameters and analyzes their monotonicity across different graph orders and colourings.

Findings

Monotone behaviour in the number of colours for two parameters

Characterizations of graphs with maximum critical set sizes

Extensions to hypergraph colouring variants

Abstract

The Sudoku number has been defined under various names, indicating it is a natural concept. There are four variants of this parameter, that can be related to the maximum and minimum size of a critical set in a graph colouring problem. For each of these four related parameters, we present some simple characterizations of the graphs attaining the maximum possible values. As a main result, we answer a question by Cooper and Kirkpatrick, showing that there is monotone behaviour in the number of colours for only two of the four parameters. We investigate the monotone behaviour for the subgraph-order as well. For Latin squares and the Sudoku, we solve some variants for hypergraph colouring.