arXiv subjects
Publications and source records attributed to Jan Kurkofka.
Carmesin has extended Robertson and Seymour's tree-of-tangles theorem to the infinite tangles of locally finite infinite graphs. We extend it further to the infinite tangles of all infinite graphs. Our result has a number of applications for the topology of infinite graphs, such as their end spaces and their compactifications.
The directions of an infinite graph $G$ are a tangle-like description of its ends: they are choice functions that choose compatibly for all finite vertex sets $X\subseteq V(G)$ a component of $G-X$. Although every direction is induced by a ray, there exist directions of graphs that are not uniquely determined by any countable subset of their choices. We characterise these directions and their countably determined counterparts in terms of star-like substructures or rays of the graph. Curiously, there exist graphs whose directions are all countably determined but which cannot be distinguished all at once by countably many choices. We structurally characterise the graphs whose directions can be distinguished all at once by countably many choices, and we structurally characterise the graphs which admit no such countably many choices. Our characterisations are phrased in terms of normal trees and tree-decompositions. Our four (sub)structural characterisations imply combinatorial characterisations of the four classes of infinite graphs that are defined by the first and second axiom of countability applied to their end spaces: the two classes of graphs whose end spaces are first countable or second countable, respectively, and the complements of these two classes.
An end of a graph $G$ is an equivalence class of rays, where two rays are equivalent if there are infinitely many vertex-disjoint paths between them in $G$. The degree of an end is the maximum cardinality of a collection of pairwise disjoint rays in this equivalence class. Halin conjectured that the end degree can be characterised in terms of certain typical ray configurations, which would generalise his famous \emph{grid theorem}. In particular, every end of regular uncountable degree $κ$ would contain a \emph{star of rays}, i.e.\ a configuration consisting of a central ray $R$ and $κ$ neighbouring rays $(R_i \colon i < κ)$ all disjoint from each other and each $R_i$ sending a family of infinitely many disjoint paths to $R$ so that paths from distinct families only meet in $R$. We show that Halin's conjecture fails for end degree $ \aleph_1$, holds for $\aleph_2,\aleph_3,\ldots,\aleph_ω$, fails for $ \aleph_{ω+1}$, and is undecidable (in ZFC) for the next $\aleph_{ω+n}$ with $n \in \mathbb{N}$, $n \geq 2$. Further results include a complete solution for all cardinals under GCH, complemented by a number of consistency results.
In a series of four papers we determine structures whose existence is dual, in the sense of complementary, to the existence of stars or combs. In the first paper of our series we determined structures that are complementary to arbitrary stars or combs. Stars and combs can be combined, positively as well as negatively. In the second and third paper of our series we provided duality theorems for all but one of the possible combinations. In this fourth and final paper of our series, we complete our solution to the problem of finding complementary structures for stars, combs, and their combinations, by presenting duality theorems for the missing piece: for undominating stars. Our duality theorems are phrased in terms of end-compactified subgraphs, tree-decompositions and tangle-distinguishing separators.
In a series of four papers we determine structures whose existence is dual, in the sense of complementary, to the existence of stars or combs. Here, in the third paper of the series, we present duality theorems for a combination of stars and combs: undominated combs. We describe their complementary structures in terms of rayless trees and of tree-decompositions. Applications include a complete characterisation, in terms of normal spanning trees, of the graphs whose rays are dominated but which have no rayless spanning tree. Only two such graphs had so far been constructed, by Seymour and Thomas and by Thomassen. As a corollary, we show that graphs with a normal spanning tree have a rayless spanning tree if and only if all their rays are dominated.
In a series of four papers we determine structures whose existence is dual, in the sense of complementary, to the existence of stars or combs. Here, in the second paper of the series, we present duality theorems for combinations of stars and combs: dominating stars and dominated combs. As dominating stars exist if and only if dominated combs do, the structures complementary to them coincide. Like for arbitrary stars and combs, our duality theorems for dominated combs (and dominating stars) are phrased in terms of normal trees or tree-decompositions. The complementary structures we provide for dominated combs unify those for stars and combs and allow us to derive our duality theorems for stars and combs from those for dominated combs. This is surprising given that our complementary structures for stars and combs are quite different: those for stars are locally finite whereas those for combs are rayless.
We show that every infinitely edge-connected graph contains the Farey graph or $T_{\aleph_0}\!\ast t$ as a minor. These two graphs are unique with this property up to minor-equivalence.
We construct a countable planar graph which, for any two vertices $u,v$ and any integer $k\ge 1$, contains $k$ edge-disjoint order-compatible $u$--$v$ paths but not infinitely many. This graph has applications in Ramsey theory, in the study of connectivity and in the characterisation of the Farey graph.
We show that the tangle space of a graph, which compactifies it, is a quotient of its Stone-Čech remainder obtained by contracting the connected components.
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. Thus, the arising tangle compactification extends the well-known Freudenthal compactification from connected locally finite graphs to arbitrary infinite graphs. This Master's thesis investigates the tangle compactifcation.
We show that an arbitrary infinite graph $G$ can be compactified by its ends plus its critical vertex sets, where a finite set $X$ of vertices of an infinite graph is critical if its deletion leaves some infinitely many components each with neighbourhood precisely equal to $X$. We further provide a concrete separation system whose $\aleph_0$-tangles are precisely the ends plus critical vertex sets. Our tangle compactification $\vert G\vert_Γ$ is a quotient of Diestel's (denoted by $\vert G\vert_Θ$), and both use tangles to compactify a graph in much the same way as the ends of a locally finite and connected graph compactify it in its Freudenthal compactification. Finally, generalising both Diestel's construction of $\vert G\vert_Θ$ and our construction of $\vert G\vert_Γ$, we show that $G$ can be compactified by every inverse limit of compactifications of the sets of components obtained by deleting a finite set of vertices. Diestel's $\vert G\vert_Θ$ is the finest such compactification, and our $\vert G\vert_Γ$ is the coarsest one. Both coincide if and only if all tangles are ends. This answers two questions of Diestel.