  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.
