Thomassen's Choosability Argument Revisited
David R. Wood, Svante Linusson · SIAM Journal on Discrete Mathematics · 2010
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 $K_5$-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.