Related papers: Forcing large complete minors in infinite graphs
Developing further Stein's recent notion of relative end degrees in infinite graphs, we investigate which degree assumptions can force a locally finite graph to contain a given finite minor, or a finite subgraph of given minimum degree.…
Bounds on the minimum degree and on the number of vertices at- taining it have been much studied for finite edge-/vertex-minimally k- connected/k-edge-connected graphs. We give an overview of the results known for finite graphs, and show…
Mader first proved that high average degree forces a given graph as a minor. Often motivated by Hadwiger's Conjecture, much research has focused on the average degree required to force a complete graph as a minor. Subsequently, various…
We prove that for any weakly convergent sequence of finite graphs with bounded vertex degrees, there exists a topological limit graphing.
We characterize the bipartite graphs that minimize the (first-degree based) entropy, among all bipartite graphs of given size, or given size and (upper bound on the) order. The extremal graphs turn out to be complete bipartite graphs, or…
A fundamental result in structural graph theory states that every graph with large average degree contains a large complete graph as a minor. We prove this result with the extra property that the minor is small with respect to the order of…
We investigate the question how `small' a graph can be, if it contains all members of a given class of locally finite graphs as subgraphs or induced subgraphs. More precisely, we give necessary and sufficient conditions for the existence of…
Huynh et al. recently showed that a countable graph $G$ which contains every countable planar graph as a subgraph must contain arbitrarily large finite complete graphs as topological minors, and an infinite complete graph as a minor. We…
There are different definitions of ends in non-locally-finite graphs which are all equivalent in the locally finite case. We prove the compactness of the end-topology that is based on the principle of removing finite sets of vertices and…
Given a class of graphs G closed under taking minors, we study the maximum degree \Delta_n of random graphs from G with n vertices. We prove several lower and upper bounds that hold with high probability. Among other results, we find…
In this paper we define a degree for ends of infinite digraphs. The well-definedness of our definition in particular resolves a problem by Zuther. Furthermore, we extend our notion of end degree to also respect, among others, the vertices…
We systematically study a natural problem in extremal graph theory, to minimize the number of edges in a graph with a fixed number of vertices, subject to a certain local condition: each vertex must be in a copy of a fixed graph $H$. We…
We study large minors in small-set expanders. More precisely, we consider graphs with $n$ vertices and the property that every set of size at most $\alpha n / t$ expands by a factor of $t$, for some (constant) $\alpha > 0$ and large $t =…
This paper is intended as an introductory survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends,…
We survey various aspects of infinite extremal graph theory and prove several new results. The lead role play the parameters connectivity and degree. This includes the end degree. Many open problems are suggested.
We extend the concept of the law of a finite graph to graphings, which are, in general, infinite graphs whose vertices are equipped with the structure of a probability space. By doing this, we obtain a vast array of new unimodular measures.…
This paper is the last part of a comprehensive survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends,…
We analyse an extremal question on the degrees of the link graphs of a finite regular graph, that is, the subgraphs induced by non-trivial spheres. We show that if $G$ is $d$-regular and connected but not complete then some link graph of…
We prove an asymptotically tight bound on the extremal density guaranteeing subdivisions of bounded-degree bipartite graphs with a mild separability condition. As corollaries, we answer several questions of Reed and Wood on embedding sparse…
In finite graphs, finite-order tangles offer an abstract description of highly connected substructures. In infinite graphs, infinite-order tangles compactify the graphs in the same way the ends compactify connected locally finite graphs.…