Minimal Connectivity

A k-connected graph such that deleting any edge / deleting any vertex / contracting any edge results in a graph which is not k-connected is called minimally / critically / contraction-critically k-connected. These three classes play a prominent role in graph connectivity theory, and we give a brief introduction with a light emphasis on reduction- and construction theorems for classes of k-connected graphs.Comment: IMADA-preprint-math, 33 page

Edge-disjoint Linkages in Infinite Graphs

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]