Hadwiger's conjecture for graphs with infinite chromatic number

We construct a connected graph H such that (1) χ(H) = ω; (2) Kω, the complete graph on ω points, is not a minor of H . Therefore Hadwiger’s conjecture does not hold for graphs with infinite coloring number. 1. Notation In this note we are only concerned with simple undirected graphs G = (V,E) where V is a set and E ⊆ P2(V ) where P2(V ) = { {x, y} : x, y ∈ V and x 6= y } . We also require that V ∩E = ∅ to avoid notational ambiguities. We denote the vertex set of a graph G by V (G) and the edge set by E(G). Moreover, for any cardinal α we denote the complete graph on α points by Kα. For any graph G, disjoint subsets S, T ⊆ V (G) are said to be connected to each other if there are s ∈ S, t ∈ T with {s, t} ∈ E(G). Note that Kα is a minor of a graph G if and only if there is a collection {Sβ : β ∈ α} of nonempty, connected and pairwise disjoint subsets of V (G) such that for all β, γ ∈ α with β 6= γ the sets Sβ and Sγ are connected to each other. We will need the following observation later on: Fact 1.1. For any graph G, finite or infinite, the following are equivalent: (1) G is connected; (2) if S, T ⊆ V (G) are nonempty and disjoint such that S ∪ T = V (G) then S, T are connected to each other. 2. The construction In [1], Hadwiger formulated his well-known and deep conjecture, linking the chromatic number χ(G) of a graph G with clique minors. His conjecture can be formulated that Kχ(G) is a minor of G for every graph G. In the following we present a connected graph H with chromatic number ω such that Kω is not a minor of H. Let N be the set of positive integers. For any n ∈ N we let Cn = {1, . . . , n} × {n} 2010 Mathematics Subject Classification. 05C15, 05C83. 1 2 DOMINIC VAN DER ZYPEN