On the complexity of the vertex $3$-coloring problem for the hereditary graph classes with forbidden subgraphs of small size

The $3$-coloring problem for a given graph consists in verifying whether it is possible to divide the vertex set of the graph into three subsets of pairwise nonadjacent vertices. A complete complexity classification is known for this problem for the hereditary classes defined by triples of forbidden induced subgraphs, each on at most $5$ vertices. In this article, the quadruples of forbidden induced subgraphs is under consideration, each on at most $5$ vertices. For all but three corresponding hereditary classes, the computational status of the $3$-coloring problem is determined. Considering two of the remaining three classes, we prove their polynomial equivalence and polynomial reducibility to the third class. Illustr. 4, bibliogr. 20.