Polynomial Cases for the Vertex Coloring Problem

TL;DR

This paper establishes polynomial-time algorithms for vertex coloring in specific hereditary graph classes defined by forbidden subgraphs, advancing the understanding of computational complexity in graph theory.

Contribution

It proves polynomial-time solvability for four previously unresolved hereditary graph classes defined by forbidden subgraphs.

Findings

Polynomial-time algorithms for ($P_5$, dart)-free graphs

Polynomial-time algorithms for ($P_5$, banner)-free graphs

Polynomial-time algorithms for ($P_5$, bull)-free graphs

Abstract

The computational complexity of the Vertex Coloring problem is known for all hereditary classes of graphs defined by forbidding two connected five-vertex induced subgraphs, except for seven cases. We prove the polynomial-time solvability of four of these problems: for ($P_5$, dart)-free graphs, ($P_5$, banner)-free graphs, ($P_5$, bull)-free graphs, and (fork, bull)-free graphs.