Colouring a graph with position sets

TL;DR

This paper introduces the g-coloring number, a new graph coloring concept based on the general position property, providing bounds, exact values for specific classes, and proving NP-completeness.

Contribution

It defines the g-coloring number, establishes bounds, computes it for various graph classes, and proves the problem is NP-complete, advancing understanding of graph coloring with geometric constraints.

Findings

Bounds on g-coloring number in terms of graph parameters

Exact g-coloring numbers for Kneser, line, multipartite, block, and Cartesian product graphs

NP-completeness of the g-coloring problem

Abstract

In this paper we consider a colouring version of the general position problem. The \emph{$\gp $-chromatic number} is the smallest number of colours needed to colour the vertices of the graph such that each colour class has the no-three-in-line property. We determine bounds on this colouring number in terms of the diameter, general position number, size, chromatic number, cochromatic number and total domination number and prove realisation results. We also determine the $\gp $-chromatic number of several graph classes, including Kneser graphs $K(n,2)$, line graphs of complete graphs, complete multipartite graphs, block graphs and Cartesian products. Finally, we show that the $\gp $-colouring problem is NP-complete.