2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/99663We 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.9 pages, appears in part in an ICTCS 2001 paper by the same authors, minor revision of the aboveComputational ComplexityF.1.3; F.2.2text