Thomassen's Choosability Argument Revisited

Thomassen (J. Combin. Theory Ser. B, 62 (1994), pp. 180–181) proved that every planar graph is 5-choosable. This result was generalized by Škrekovski (Discrete Math., 190 (1998), pp. 223–226) and He, Miao, and Shen (Discrete Math., 308 (2008), pp. 4024–4026), who proved that every K5-minor-free graph is 5-choosable. Both proofs rely on the characterization of K5-minorfree graphs due to Wagner (Math. Ann., 114 (1937), pp. 570–590). This paper proves the same result without using Wagner’s structure theorem or even planar embeddings. Given that there is no structure theorem for graphs with no K6-minor, we argue that this proof suggests a possible approach for attacking the Hadwiger Conjecture.