4-coloring ($P_6$, bull)-free graphs

TL;DR

This paper introduces polynomial-time algorithms for 4-coloring ($P_6$, bull)-free graphs and extends to k-coloring in ($P_6$, bull, gem)-free graphs, advancing graph coloring theory for specific graph classes.

Contribution

It provides the first polynomial-time algorithms for 4-coloring ($P_6$, bull)-free graphs and generalizes to k-coloring in ($P_6$, bull, gem)-free graphs.

Findings

Polynomial-time 4-coloring algorithm for ($P_6$, bull)-free graphs

Polynomial-time k-coloring algorithm for ($P_6$, bull, gem)-free graphs

Extension of coloring algorithms to broader graph classes

Abstract

We present a polynomial-time algorithm that determines whether a graph that contains no induced path on six vertices and no bull (the graph with vertices a, b, c, d, e and edges ab, bc, cd, be, ce) is 4-colorable. We also show that for any fixed k the k-coloring problem can be solved in polynomial time in the class of ($P_6$, bull, gem)-free graphs.