Forcing highly connected subgraphs

Abstract

A theorem of Mader states that highly connected subgraphs can be forced in finite graphs by assuming a high minimum degree. We extend this result to infinite graphs. Here, it is necessary to require not only high degree for the vertices but also high vertex-degree (or multiplicity) for the ends of the graph, that is, a large number of disjoint rays in each end.We give a lower bound on the degree of vertices and the vertex-degree of the ends which is quadratic in k, the connectedness of the desired subgraph. In fact, this is not far from best possible: we exhibit a family of graphs with a degree of order 2 k at the vertices and a vertex-degree of order k log k at the ends which have no k-connected subgraphs.Furthermore, if in addition to the high degrees at the vertices, we only require high edge-degree for the ends (which is defined as the maximum number of edge-disjoint rays in an end), Mader's theorem does not extend to infinite graphs, not even to locally finite ones. We give a counterexample in this respect. But, assuming a lower bound of at least 2k for the edgedegree at the ends and the degree at the vertices does suffice to ensure the existence (k + 1)-edge-connected subgraphs in arbitrary graphs.

Forcing Finite Minors in Sparse Infinite Graphs by Large-Degree Assumptions

“…An assumption of large end degrees in this sense, as well as of large vertex degrees, can force some substructures in locally finite infinite graphs, such as an unspecified highly connected subgraph [3,4,5,6]. However, it will not yield the kind of result we are seeking here: for every integer k there are planar graphs with all vertex and end degrees at least k, and hence no such degree assumption can force even a K 5 minor.…”

Forcing Large Complete (Topological) Minors in Infinite Graphs

“…This approach has also proved successful in other recent work [4,14,12]. In this way, that is, defining the degree of an end in an appropriate way, the minimum degree, now taken over vertices and ends, can continue to serve as our condition for forcing large complete minors.…”

Ends and Vertices of Small Degree in Infinite Minimally k-(Edge)-Connected Graphs

“…We also need the following lemma from [27]. Combined, the two lemmas yield a lemma similar to Lemma 7 from the previous section: Lemma 11.…”

“…The concept of the end degree has been introduced in [4] and [27], see also [8]. In fact we distiguish between two types of end degrees: the vertex-degree and the edge-degree.…”