On vertex coloring without monochromatic triangles

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 χ3. We bound χ3 by other known structural parameters. We also present two classes of graphs with interesting coloring properties, that play pivotal role in proving useful observations about our problem.