The coloring problem for $\{P_5,\bar{P_5}\}$-free graphs and $\{P_5,K_p-e\}$-free graphs is polynomial

TL;DR

This paper proves that the problem of finding the chromatic number in certain classes of graphs, specifically those free of specific induced paths and complements, can be solved efficiently using polynomial algorithms.

Contribution

It establishes polynomial-time algorithms for the chromatic number problem in $ ext{P}_5$-free graphs with additional constraints, expanding the classes of graphs with tractable coloring problems.

Findings

Chromatic number determination is polynomial for $ ext{P}_5,ar{ ext{P}_5}$-free graphs.

Chromatic number determination is polynomial for $ ext{P}_5,K_p-e$-free graphs.

The results identify new tractable cases in graph coloring complexity.

Abstract

We show that determining the chromatic number of a $\{P_5,\bar{P_5}\}$-free graph or a $\{P_5,K_p-e\}$-free graph can be done in polynomial time