Planar Graph Coloring is not Self-Reducible, Assuming P != NP

We show that obtaining the lexicographically first four coloring of a planar graph is NP–hard. This shows that planar graph four-coloring is not self-reducible, assuming P 6= NP . One consequence of our result is that the schema of [JVV 86] cannot be used for approximately counting the number of four colorings of a planar graph. These results extend to planar graph k-coloring, for k ≥ 4. Research done while this author was supported by an IBM Graduate Fellowship at Cornell University. Work done while at Cornell University. Supported by NSF grant DCR 85-52938 and PYI matching funds from AT&T Bell Labs and Sun Microsystems, Inc.