| 11. | All other points remain # P-hard, even for bipartite planar graphs.
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| 12. | In the non-Hamiltonian maximal planar graph.
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| 13. | Thus, a planar graph has thickness 1.
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| 14. | Planar graphs are graphs embedded into a plane.
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| 15. | The intuitive idea underlying discharging is to consider the planar graph as an electrical network.
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| 16. | If this were the restriction, planar graphs would require arbitrarily large numbers of colors.
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| 17. | This allows drawing methods for planar graphs to be extended to non-planar graphs.
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| 18. | This allows drawing methods for planar graphs to be extended to non-planar graphs.
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| 19. | For instance, the 16-vertex planar graph shown in the illustration has edges.
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| 20. | For planar graphs with maximum degree, the optimal number of colors is again exactly.
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