Graphs with Sudoku number $n-1$

TL;DR

This paper proves that the only connected graphs with Sudoku number n-1 are complete graphs, confirming a conjecture about the structure of Sudoku colourings in graph theory.

Contribution

The paper confirms the conjecture that connected graphs with Sudoku number n-1 are exactly the complete graphs, providing a complete characterization.

Findings

Connected graphs with Sudoku number n-1 are complete graphs.

The conjecture by Lau-Jeyaseeli-Shiu-Arumugam is proven true.

The result characterizes the structure of Sudoku colourings in graphs.

Abstract

Recently Lau-Jeyaseeli-Shiu-Arumugam introduced the concept of the "Sudoku colourings" of graphs -- partial $\chi(G)$-colourings of $G$ that have a unique extension to a proper $\chi(G)$-colouring of all the vertices. They introduced the Sudoku number of a graph as the minimal number of coloured vertices in a Sudoku colouring. They conjectured that a connected graph has Sudoku number $n-1$ if, and only if, it is complete. In this note we prove that this is true.