On computing the smallest four-coloring of planar graphs and non-self-reducible sets in P

We show that computing the lexicographically first four-coloring for planar graphs is -hard. This result optimally improves upon a result of Khuller and Vazirani who prove this problem NP-hard, and conclude that it is not self-reducible in the sense of Schnorr, assuming P≠NP. We discuss this application to non-self-reducibility and provide a general related result. We also discuss when raising a problem's NP-hardness lower bound to -hardness can be valuable.