Coloring ($P_5$, bull)-free graphs

TL;DR

This paper presents a polynomial-time algorithm for determining the chromatic number of graphs that exclude both a path on five vertices and a specific five-vertex graph called the bull as induced subgraphs.

Contribution

It introduces a novel polynomial-time algorithm for coloring ($P_5$, bull)-free graphs, expanding the class of graphs with efficiently computable chromatic numbers.

Findings

Algorithm computes chromatic number in polynomial time

Applicable to ($P_5$, bull)-free graphs

Advances understanding of graph coloring complexity

Abstract

We give a polynomial-time algorithm that computes the chromatic number of any graph that contains no path on five vertices and no bull as an induced subgraph (where the bull is the graph with five vertices $a,b,c,d,e$ and edges $ab,bc,cd,be,ce$).