Balanced Black and White Coloring Problem on knights chessboards

TL;DR

This paper investigates a complex graph anticoloring problem involving placing black and white knights on a chessboard so that no attacking queens of different colors are adjacent, focusing on the challenging balanced case.

Contribution

It introduces the balanced black and white coloring problem specifically for knights on chessboards, highlighting its computational difficulty and open complexity status.

Findings

Problem remains computationally challenging.

Focus on knight piece due to complexity.

Open questions about problem complexity.

Abstract

Graph anticoloring problem is partial coloring problem where the main feature is the opposite rule of the graph coloring problem, i.e., if two vertices are adjacent, their assigned colors must be the same or at least one of them is uncolored. In the same way, Berge in 1972 proposed the problem of placing b black queens and w white queens on a $n \times n$ chessboard such that no two queens of different color can attack to each other, the complexity of this problem remains open. In this work we deal with the knight piece under the balance property, since this special case is the most difficult for brute force algorithms.