Edge-disjoint Linkages in Infinite Graphs

Abstract

The main subject of this thesis is the infinite graph version of the weak linkage conjecture by Thomassen [24]. We first prove results about the structure of the lifting graph; Theorems 2.2.8, 2.2.24, and 2.3.1. As an application, we improve the weak-linkage result of Ok, Richter, and Thomassen [18]. We show that an edge-connectivity of (k+1) is enough to have a weak k-linkage in infinite graphs in case k is odd, Theorem 3.3.6. Thus proving that Huck's theorem holds for infinite graphs. This is only one step far away from the conjecture, which has an edge-connectivity condition of only k in case k is odd. As another application, in Theorem 4.2.7 we improve a result of Thomassen about strongly connected orientations of infinite graphs [25], in the case when the infinite graph is 1-ended. This brings us closer to proving the orientation conjecture of Nash-Williams for infinite graphs [15].