Polynomial Cases for the Vertex Coloring Problem
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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.
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