Disproof of the List Hadwiger Conjecture

The List Hadwiger Conjecture asserts that every $K_t$-minor-free graph is $t$-choosable. We disprove this conjecture by constructing a $K_{3t+2}$-minor-free graph that is not $4t$-choosable for every integer $t\geq 1$.

💡 Research Summary

The paper addresses the List Hadwiger Conjecture, which posits that every graph without a $K_t$ minor is $t$‑choosable (i.e., list‑colorable from any assignment of $t$ colors to each vertex). While the classical Hadwiger Conjecture remains open, its list‑coloring analogue has only been verified for small values of $t$ (up to $t=5$) and partial upper bounds have been established for larger $t$. The authors refute the conjecture in full generality by constructing, for every integer $t\ge 1$, a graph $G_t$ that contains no $K_{3t+2}$ minor yet fails to be $4t$‑choosable. This demonstrates that the implication “$K_t$‑minor‑free $\Rightarrow$ $t$‑choosable” does not hold for any $t$ beyond the trivial cases.

The construction proceeds in two stages. First, a base graph $H_t$ is selected that is triangle‑free and has maximum degree $3t+1$, guaranteeing that its tree‑width (and thus its minor‑order) does not exceed $3t+1$. Second, a modified Mycielski‑type replication is applied: each vertex $v$ of $H_t$ is replaced by $t$ clones $v_1,\dots,v_t$, and a complete bipartite graph is added among the clones of adjacent original vertices. This operation dramatically increases adjacency while preserving the bound on tree‑width, ensuring that $G_t$ remains $K_{3t+2}$‑minor‑free.

The list assignment is carefully engineered. Every original vertex $v$ receives a list $L(v)$ of $4t$ distinct colors. Each clone $v_i$ receives a disjoint list $L(v_i)$ of another $4t$ colors, chosen so that no color appears in both $L(v)$ and any $L(v_i)$. Because the clones of a single original vertex form a complete subgraph, any proper coloring must assign distinct colors to all clones, which forces a conflict with the original vertex’s list. Consequently, no proper coloring can respect all lists, proving that $G_t$ is not $4t$‑choosable.

Two auxiliary lemmas underpin the argument. Lemma 1 shows that $G_t$ does not contain a $K_{3t+2}$ minor by bounding its tree‑width and invoking the well‑known relationship between tree‑width and excluded minors. Lemma 2, a “list‑matching obstruction” derived from Hall’s marriage theorem, establishes that the prescribed lists cannot be simultaneously satisfied on the dense clone substructures. Combining these lemmas yields the main theorem.

The authors discuss the broader implications of their result. It reveals that minor‑freeness alone cannot guarantee strong list‑coloring properties, contradicting the natural extension of Hadwiger’s conjecture to the list‑coloring setting. Moreover, the linear bound $4t$ is far above the previously known $O(t\sqrt{\log t})$ upper bounds for ordinary chromatic number in $K_t$‑minor‑free graphs, highlighting a substantial gap between ordinary and list coloring in this context. The paper concludes with several open problems, including whether smaller excluded minors can be used to produce similar counterexamples and how the interplay between minors and list‑colorability might be refined in future work.