Forcing highly connected subgraphs

TLDR: In this paper, 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 is given.

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 2k 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 edge-degree at the ends and the degree at the vertices does suffice to ensure the existence (k + 1)-edge-connected subgraphs in arbitrary graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 331–349, 2007