Polynomial Cases for the Vertex Coloring Problem

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.