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 Skrekovski (Discrete Math., 190 (1998), pp. 223-226) and He, Miao, and Shen (Discrete Math., 308 (2008), pp. 4024-4026), who proved that every $K_5$-minor-free graph is 5-choosable. Both proofs rely on the characterization of $K_5$-minor-free 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 $K_6$-minor, we argue that this proof suggests a possible approach for attacking the Hadwiger Conjecture.

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