On vertex coloring without monochromatic triangles

TL;DR

This paper investigates a relaxed vertex coloring problem avoiding monochromatic triangles, providing a classification, complexity results, and introducing the triangle-free chromatic number as a new structural parameter.

Contribution

It introduces the triangle-free chromatic number, offers a classification of the problem, and improves computational complexity bounds for coloring without monochromatic triangles.

Findings

Bounded the triangle-free chromatic number by known parameters

Classified the problem within classic and parametrized algorithms

Presented graph classes with unique coloring properties

Abstract

We study a certain relaxation of the classic vertex coloring problem, namely, a coloring of vertices of undirected, simple graphs, such that there are no monochromatic triangles. We give the first classification of the problem in terms of classic and parametrized algorithms. Several computational complexity results are also presented, which improve on the previous results found in the literature. We propose the new structural parameter for undirected, simple graphs -- the triangle-free chromatic number $\chi_3$. We bound $\chi_3$ by other known structural parameters. We also present two classes of graphs with interesting coloring properties, that play pivotal role in proving useful observation about our problem. We give/ask several conjectures/questions throughout this paper to encourage new research in the area of graph coloring.