A Note on the Complexity of Computing the Smallest Four-Coloring of Planar Graphs
<|reference_start|>A Note on the Complexity of Computing the Smallest Four-Coloring of Planar Graphs: We show that computing the lexicographically first four-coloring for planar graphs is P^{NP}-hard. This result optimally improves upon a result of Khuller and Vazirani who prove this problem to be NP-hard, and conclude that it is not self-reducible in the sense of Schnorr, assuming P \neq NP. We discuss this application to non-self-reducibility and provide a general related result.<|reference_end|>
@article{grosse2001a,
    title={A Note on the Complexity of Computing the Smallest Four-Coloring of
  Planar Graphs},
    author={Andre Grosse, Joerg Rothe, and Gerd Wechsung},
    journal={arXiv preprint arXiv:cs/0106045},
    year={2001},
    archivePrefix={arXiv},
    eprint={cs/0106045},
    primaryClass={cs.CC}
}