Disproof of the List Hadwiger Conjecture

Hadwiger Conjecture. Every K t -minor-free graph is (t -1)-colourable.

The Hadwiger Conjecture holds for t ≤ 6 (see [3,6,17,18,28]) and is open for t ≥ 7. In fact, the following more general conjecture is open.

Weak Hadwiger Conjecture. Every K t -minor-free graph is ct-colourable for some constant c ≥ 1.

It is natural to consider analogous conjectures for list colourings 2 . First, consider the choosability of planar graphs. Erdős et al. [5] conjectured that some planar graph is not 4-choosable, and that every planar graph is 5-choosable. The first conjecture was verified by Voigt [27] and the second by Thomassen [25]. Incidentally, Borowiecki [1] asked whether every K t -minor-free graph is (t -1)-choosable, which is true for t ≤ 4 but false for t = 5 by Voigt’s example. The following natural conjecture arises (see [10,30]).

List Hadwiger Conjecture. Every K t -minor-free graph is t-choosable.

The List Hadwiger Conjecture holds for t ≤ 5 (see [7,20,31]). Again the following more general conjecture is open.

Weak List Hadwiger Conjecture. Every K t -minor-free graph is ct-choosable for some constant c ≥ 1.

In this paper we disprove the List Hadwiger Conjecture for t ≥ 8, and prove that c ≥ 4 3 in the Weak List Hadwiger Conjecture.

Theorem 1. For every integer t ≥ 1, (a) there is a K 3t+2 -minor-free graph that is not 4t-choosable. (b) there is a K 3t+1 -minor-free graph that is not (4t -2)-choosable, (c) there is a K 3t -minor-free graph that is not (4t -3)-choosable.

Before proving Theorem 1, note that adding a dominant vertex to a graph does not necessarily increase the choice number (as it does for the chromatic number). For example, K 3,3 is 3-choosable but not 2-choosable. Adding one dominant vertex to K 3,3 gives K 1,3,3 , which again is 3-choosable [16]. In fact, this property holds for infinitely many complete bipartite graphs [16]; also see [19].

Let G 1 and G 2 be graphs, and let S i be a k-clique in each G i . Let G be a graph obtained from the disjoint union of G 1 and G 2 by pairing the vertices in S 1 and S 2 and identifying each pair. Then G is said to be obtained by pasting G 1 and G 2 on S 1 and S 2 . The following lemma is well known. Lemma 2. Let G 1 and G 2 be K t -minor-free graphs. Let S i be a k-clique in each G i . Let G be a pasting of G 1 and G 2 on S 1 and S 2 . Then G is K t -minor-free.

Proof. Suppose on the contrary that K t+1 is a minor of G. Let X 1 , . . . , X t+1 be the corresponding branch sets. If some X i does not intersect G 1 and some X j does not intersect G 2 , then no edge joins X i and X j , which is a contradiction. Thus, without loss of generality, each

1 , . . . , X ′ t+1 are the branch sets of a K t+1 -minor in G 1 . This contradiction proves that G is K t -minor-free.

Let K r×2 be the complete r-partite graph with r colour classes of size 2. Let K 1,r×2 be the complete (r + 1)-partite graph with r colour classes of size 2 and one colour class of size 1. That is, K r×2 and K 1,r×2 are respectively obtained from K 2r and K 2r+1 by deleting a matching of r edges. The following lemma will be useful. Lemma 3 ([8,29]). K r×2 is K ⌊3r/2⌋+1 -minor-free, and K 1,r×2 is K ⌊3r/2⌋+2 -minor-free.

Proof of Theorem 1. Our goal is to construct a K p -minor-free graph and a non-achievable list assignment with q colours per vertex, where the integers p, q and r and a graph H are defined in the following table. Let {v 1 w 1 , . . . , v r w r } be the deleted matching in H. By Lemma 3, the calculation in the table shows that H is K p -minor-free.

For each vector (c 1 , . . . , c r ) ∈ [1, q] r , let H(c 1 , . . . , c r ) be a copy of H with the following list assignment.

for each remaining vertex u. There are q + 1 colours in total, and |V (H)| = q + 2. Thus in every L-colouring of H, two non-adjacent vertices receive the same colour. That is, col

Let G be the graph obtained by pasting all the graphs H(c 1 , . . . , c r ), where (c 1 , . . . , c r ) ∈ [1, q] r , on the clique {v 1 , . . . , v r }. The list assignment L is well defined for G since L(v i ) = [1, q]. By Lemma 2, G is K p -minor-free. Suppose that G is L-colourable. Let c i be the colour assigned to each vertex v i . Thus c i ∈ L(v i ) = [1, q]. Hence, as proved above, the copy H(c 1 , . . . , c r ) is not L-colourable. This contradiction proves that G is not L-colourable. Each vertex of G has a list of q colours in L. Therefore G is not q-choosable. (It is easily seen that G is q-degenerate3 , implying G is (q + 1)-choosable.) Note that this proof was inspired by the construction of a non-4-choosable planar graph by Mirzakhani [15].

Theorem 1 disproves the List Hadwiger Conjecture. However, list colourings remain a viable approach for attacking Hadwiger’s Conjecture. Indeed, list colourings provide potential routes around some of t