The Hadwiger number of infinite vertex-transitive graphs

We prove that every infinite, locally finite 3-connected, almost 4-connected, almost transitive, nonplanar graph, which contains infinitely many pairwise disjoint infinite paths belonging to the same end, can be contracted into an infinite complete graph. This implies that every infinite, locally finite, connected, nonplanar vertex-transitive graph with only one end can be contracted into an infinite complete graph. This problem was raised by L. Babai.