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

Authors:
André Große;Jörg Rothe;Gerd Wechsung
Affiliations:
Entwicklung Lasersensorik, JENOP-TIK Laser, Optik, Systeme GmbH, Jena, Germany and Institut für Informatik, Friedrich-Schiller-Universität Jena, Jena, Germany;Institut für Informatik, Heinrich-Heine-Universität Düsseldorf, Düsseldorf, Germany;Institut für Informatik, Friedrich-Schiller-Universität Jena, Jena, Germany
Venue:
Information Processing Letters
Year:
2006

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Abstract

We show that computing the lexicographically first four-coloring for planar graphs is Δ2p-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 Δ2p-hardness can be valuable.