We show that obtaining the lexicographically first four coloring of a planar graph is NP-hard. This shows that planar graph four-coloring is not self-reducible, assuming P≠ NP. One consequence of our result is that the schema of Jerrum et al. (1986) cannot be used for approximately counting the number of four colorings of a planar graph. These results extend to planar graph k-coloring, for k≥4. © 1991.
Khuller, S., & Vazirani, V. V. (1991). Planar graph coloring is not self-reducible, assuming P ≠ NP. Theoretical Computer Science, 88(1), 183–189. https://doi.org/10.1016/0304-3975(91)90081-C