On the tangle compactification of infinite graphs

作为缠结的端点

Ends as tangles

本文研究无限图端点所诱导的缠结，刻画了各有限阶闭缠结的特征，并给出其与多数投票顶点集及顶点度数、支配顶点数之间的等价条件。

极限封闭轮廓

Limit-closed Profiles

本文针对无限分离系统提出了一个缠结树定理，其特点是不对分离系统本身施加限制，而是对要在树中排列的缠结施加限制。

Pseudoflowers in infinite connectivity systems

Given a graph or a matroid, a tree of tangles is a tree decomposition that displays the structure of the connectivity: every edge of the decomposition tree induces a separation, that is, a way to divide the graph or matroid into two parts; and for every two highly connected areas (encoded as tangles) that live on different sides of some separation, some separation induced by an edge distinguishes them. Separations induced by a tree of tangles cannot cross. One approach to display even more connectivity structure is to insert even more structure into a tree of tangles, for example the flowers t…

通过布尔代数和函子描述端和缠结（及其边变体）

Describing ends and tangles (and their edge variants) through Boolean algebras and functors

本文通过布尔代数和Stone对偶，统一了无限图的端空间、缠结空间和边方向空间，并给出了缠结的边类比定义，证明了它们等价于边方向。

Duality theorems for stars and combs I: Arbitrary stars and combs

Extending the well-known star-comb lemma for infinite graphs, we characterise the graphs that do not contain an infinite comb or an infinite star, respectively, attached to a given set of vertices. We offer several characterisations: in terms of normal trees, tree-decompositions, ranks of rayless graphs and tangle-distinguishing separators.

Fair division of graphs and of tangled cakes

A tangle is a connected topological space constructed by gluing several copies of the unit interval $[0, 1]$. We explore which tangles guarantee envy-free allocations of connected shares for n agents, meaning that such allocations exist no matter which monotonic and continuous functions represent agents' valuations. Each single tangle $\mathcal{T}$ corresponds in a natural way to an infinite topological class $\mathcal{G}(\mathcal{T})$ of multigraphs, many of which are graphs. This correspondence links EF fair division of tangles to EFk$_{outer}$ fair division of graphs. We know from Bilò et a…

End compactifications in non-locally-finite graphs

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 give a proof of the compactness of the end-topology that is constructed by the principle of removing finite sets of edges. For the latter case there exists already a proof in \cite{cartwright93martin}, which only works on graphs with countably infinite vertex sets and in contrast to which we do not use the Theorem of Tychonoff. We also construct a new topology…

Profinite分离系统

Profinite separation systems

本文研究无限分离系统与其诱导的有限分离系统之间的关系，建立profinite分离系统的基础理论，为后续证明无限图和拟阵的tangle型对偶定理奠定框架。

通过局部分离实现规范图分解

Canonical graph decompositions via local separations

本文提出通过局部分离理论实现图的规范分解，解决了传统方法中无限覆盖图不可计算的问题，并证明了DJKK26中的分解可计算性。

From cycles to circles in Cayley graphs

For locally finite infinite graphs the notion of Hamilton cycles can be extended to Hamilton circles, homeomorphic images of $S^1$ in the Freudenthal compactification. In this paper we extend some well-known theorems of the Hamiltonicity of finite Cayley graphs to infinite Cayley graphs.

An analogue of Edmonds' Branching Theorem for infinite digraphs

We extend Edmonds' Branching Theorem to locally finite infinite digraphs. As examples of Oxley or Aharoni and Thomassen show, this cannot be done using ordinary arborescences, whose underlying graphs are trees. Instead we introduce the notion of pseudo-arborescences and prove a corresponding packing result. Finally, we verify some tree-like properties for these objects, but give also an example that their underlying graphs do in general not correspond to topological trees in the Freudenthal compactification of the underlying multigraph of the digraph.

Faithful subgraphs and Hamiltonian circles of infinite graphs

A circle of an infinite locally finite graph $G$ is the imagine of a homeomorphic mapping of the unit circle $S^1$ in $|G|$, the Freudenthal compactification of $G$. A circle of $G$ is Hamiltonian if it meets every vertex (and then every end) of $G$. In this paper, we study a method for finding Hamiltonian circles of graphs. We illustrate this by extending several results on finite graphs to Hamiltonian circles in infinite graphs. For example, we prove that the prism of every 3-connected cubic graph has a Hamiltonian circle, extending the result of the finite case by Paulraja.

Hamilton circles in Cayley graphs

For locally finite infinite graphs the notion of Hamilton cycles can be extended to Hamilton circles, homeomorphic images of $S^1$ in the Freudenthal compactification. In this paper we prove of a sufficient condition for the existence of Hamilton circles in locally finite Cayley graphs.

The maximal chains of the extended Bruhat orders on the (W x W)-orbits of an infinite Renner monoid

Let (W, S) be a Coxeter system. We investigate combinatorially certain partial orders, called extended Bruhat orders, on a (W x W)-set W(N,C), which depends on W, a subset N of S, and a component C of N. We determine the length of the maximal chains between two elements. These posets generalize W equipped with its Bruhat order. They include the (W x W)-orbits of the Renner monoids of reductive algebraic monoids and of some infinite dimensional generalizations which are equipped with the partial orders obtained by the closure relations of the Bruhat and Birkhoff cells. They also include the (W …

A sufficient condition for Hamiltonicity in locally finite graphs

Using topological circles in the Freudenthal compactification of a graph as infinite cycles, we extend to locally finite graphs a result of Oberly and Sumner on the Hamiltonicity of finite graphs. This answers a question of Stein, and gives a sufficient condition for Hamiltonicity in locally finite graphs.