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

Perhaps the most basic question in extremal graph theory asks which average degree assumptions force a finite graph G to contain a given desired substructure. When this substructure is not a subgraph, but a given minor or topological minor H, its presence can be forced by making the average degree of G large enough in terms of H only, independently of the order of G. When G is infinite, this will no longer work: since infinite trees can have arbitrarily large degrees, no minimum degree assumption can force an infinite graph to contain even a cycle. However, just as a finite tree with internal vertices of large degree has many leaves, an infinite tree of large minimum degree has many ends. The question we pursue in this paper is which notion of ‘degree’ for ends might imply that an infinite graph whose vertices and ends have large degree contains a given finite minor. (The ends of a tree should then have degree 1, so that trees are no longer counterexamples.) Various notions of end degrees already exist, but only one of them, due to Stein [5], can achieve this aim. While this was a crucial step forward, the exact notion of the ‘relative’ end degrees she proposed still leaves room for improvement: it can be hard to verify that a given graph has large end degrees in this sense, indeed there seem to be only few sparse graphs that do. Our aim in this paper is to develop the notion of relative end degree further, so that more graphs satisfy the premise that both their vertices and their ends have degree at least some given k 2 N, while keeping the notion strong enough that large vertex and end degrees can force any desired finite minor. To indicate the notion of end degree that we have in mind, let us look at a locally finite connected graph G of minimum degree k which, however, has no finite subgraph of minimum degree at least k. (Such a subgraph would be good enough for us: recall that for every finite graph H there exists an integer k such that every finite graph of minimum degree at least k contains H as a minor.) If G has a finite set S of vertices such that every component C of G S is such that all its vertices with a neighbour in S have degree at least k in the