Thomassens Choosability Argument Revisited

Thomassen (1994) proved that every planar graph is 5-choosable. This result was generalised by {\v{S}}krekovski (1998) and He et al. (2008), who proved that every $K_5$-minor-free graph is 5-choosable. Both proofs rely on the characterisation of $K_5$-minor-free graphs due to Wagner (1937). 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.

💡 Research Summary

The paper revisits the classic result that every planar graph is 5‑choosable, originally proved by Thomassen, and its extension to all K₅‑minor‑free graphs proved by Škrekovski and later by He et al. Those earlier proofs crucially rely on Wagner’s structural theorem for K₅‑minor‑free graphs, which essentially reduces the problem to planar embeddings. Wood and Linusson present a completely different proof that avoids any use of Wagner’s theorem or planar embeddings.

The key innovation is the introduction of the M‑boundary concept. For a minor‑closed class M (here M is the class of K₅‑minor‑free graphs), a vertex set B⊆V(G) is an M‑boundary of G if, after adding a new vertex α adjacent to every vertex of B, the resulting graph still belongs to M. In a planar graph this exactly captures the set of vertices on the outer face, but the definition works for any graph in a minor‑closed class, thus providing a purely combinatorial analogue of the outer‑face condition used in Thomassen’s argument.

Several auxiliary lemmas are proved. Lemma 3 shows that if v∈B, then after deleting v and adding its neighbours to the boundary, the new set remains an M‑boundary. Lemma 4 (a short proof of a known Mader result) guarantees that in any 2‑connected graph one can contract an incident edge while preserving 2‑connectivity. Lemmas 5 and 6 develop the “good/bad” vertex terminology and prove that a graph contains a K₃‑minor rooted at any three distinct vertices iff every vertex is “good”; consequently, a graph is 2‑connected exactly when it contains every rooted K₃‑minor.

The main technical statement (Lemma 7, which directly yields Theorem 1) is a list‑colouring extension: given a K₅‑minor‑free graph G, a clique A⊆B⊆V(G) with |L(x)|=1 for x∈A (and distinct singleton lists), |L(x)|≥3 for x∈B\A, and |L(x)|≥5 for the remaining vertices, then G is L‑colourable. The proof proceeds by assuming a minimal counterexample and performing a detailed case analysis (nine cases). The cases handle situations where B or A is empty, G is disconnected, G has cut‑vertices, cut‑sets of size two, vertices of B with degree at least three inside B, and the relationship between the induced subgraphs G