The coloring problem for classes with two small obstructions

TL;DR

This paper investigates the graph coloring problem within classes characterized by two small forbidden induced subgraphs, providing conditions for solvability and classifying complexity for most such classes with up to five vertices.

Contribution

It establishes sufficient conditions for solving the coloring problem in these classes and determines the complexity for nearly all relevant cases with small forbidden subgraphs.

Findings

Identifies conditions for effective coloring in classes with two small obstructions.

Classifies the computational complexity for most classes with two forbidden subgraphs of up to five vertices.

Explicitly enumerates 13 cases where complexity remains unresolved.

Abstract

The coloring problem is studied in the paper for graph classes defined by two small forbidden induced subgraphs. We prove some sufficient conditions for effective solvability of the problem in such classes. As their corollary we determine the computational complexity for all sets of two connected forbidden induced subgraphs with at most five vertices except 13 explicitly enumerated cases