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Jan Kurkofka

Publications and source records attributed to Jan Kurkofka.

At least 19 recordsLinked to original sources
arXiv2026

Canonical graph decompositions via local separations

Every finite graph $G$ can be decomposed in a canonical way that displays its local connectivity-structure [DJKK26]. These decompositions are defined via a suitable more tree-like covering of $G$, whose tangle-tree structure is projected down to $G$. The covering graphs needed here are almost always infinite, and their tangle-tree structure is defined in terms of their (global) low-order separations. The canonical decompositions they induce on $G$ are therefore not computable following their definition. We reconstruct these decompositions of $G$ from finite information in $G$ itself that is sufficiently local to be reflected in the cover. This involves the reconstruction of canonical tangle structure in terms of a new theory of local separations in finite graphs, which we develop for this purpose. As an application, we find that the canonical graph-decompositions from [DJKK26] are computable.

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arXiv2026

A Tutte-type canonical decomposition of 3- and 4-connected graphs

We provide a unique decomposition of every 4-connected graph into parts that are either quasi-5-connected, cycles of triangle-torsos and 3-connected torsos on $\leq 5$ vertices, generalised double-wheels, or thickened $K_{4,m}$'s. The decomposition can be described in terms of a tree-decomposition but with edges allowed in the adhesion-sets. Our construction is explicit, canonical, and exhibits a defining property of the Tutte-decomposition. As a corollary, we obtain a new Tutte-type canonical decomposition of 3-connected graphs into parts that are either quasi-4-connected, generalised wheels or thickened $K_{3,m}$'s. This decomposition is similar yet different from the tri-separation decomposition. As an application of the decomposition for 4-connectivity, in a follow-up paper we obtain a new theorem characterising all vertex-transitive finite connected graphs as essentially quasi-5-connected or on a short explicit list of graphs.

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arXiv2025

Canonical Decompositions of 3-Connected Graphs

We offer a new structural basis for the theory of 3-connected graphs, providing a unique decomposition of every such graph into parts that are either quasi 4-connected, wheels, or thickened $K_{3,m}$'s. Our construction is explicit, canonical, and has the following applications: we obtain a new theorem characterising all finite Cayley graphs as either essentially 4-connected, cycles, or complete graphs on at most four vertices, and we provide an automatic proof of Tutte's wheel theorem.

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arXiv2025

Canonical graph decompositions via coverings

We present a canonical way to decompose finite graphs into highly connected local parts. The decomposition depends only on an integer parameter whose choice sets the intended degree of locality. The global structure of the graph, as determined by the relative position of these parts, is described by a coarser $\it model$. This is a simpler graph determined entirely by the decomposition, not imposed. The model and decomposition are obtained as projections of the tangle-tree structure of a covering of the given graph that reflects its local structure while unfolding its global structure. In this way, the tangle theory from graph minors is brought to bear canonically on arbitrary graphs, which need not be tree-like. Our theorem extends to locally finite quasi-transitive graphs, and in particular to locally finite Cayley graphs. It thereby offers a canonical decomposition for finitely generated groups into local parts, whose relative structure is displayed by a graph.

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arXiv2025

Characterising 4-tangles through a connectivity property

Every large $k$-connected graph-minor induces a $k$-tangle in its ambient graph. The converse holds for $k\le 3$, but fails for $k\ge 4$. This raises the question whether `$k$-connected' can be relaxed to obtain a characterisation of $k$-tangles through highly cohesive graph-minors. We show that this can be achieved for $k=4$ by proving that internally 4-connected graphs have unique 4-tangles, and that every graph with a 4-tangle $τ$ has an internally 4-connected minor whose unique 4-tangle lifts to $τ$.

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arXiv2025

Towards a Stallings-type theorem for finite groups

A recent development in graph-minor theory is to study local separators, vertex-sets that separate graphs locally but not necessarily globally. The local separators of a graph roughly correspond to the genuine separators of its local covering: a usually infinite graph obtained by keeping all local structure of the original graph while unfolding all other structure as much as possible. We use local separators and local coverings to discover and prove a low-order Stallings-type result for finite nilpotent groups $Γ$: the $r$-local covering of some Cayley graph $G$ of $Γ$ has $\geq 2$ ends that are separated by $\leq 2$ vertices iff $G$ has an $r$-local separator of size $\leq 2$ and $Γ$ has order $>r$, iff $Γ$ is isomorphic to $C_i\times C_j$ for some $i>r$ and $j\in\{1,2\}$.

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arXiv2024

On the edge-chromatic number of 2-complexes

We propose an open question that seeks to generalise the Four Colour Theorem from two to three dimensions. As an appetiser, we show that 12 instead of four colours are both sufficient and necessary to colour every 2-complex that embeds in a prescribed 3-manifold. However, our example of a 2-complex that requires 12 colours is not simplicial.

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arXiv2024

A representation theorem for end spaces of infinite graphs

End-spaces of infinite graphs naturally generalise the Freudenthal boundary and sit at the interface between graph theory, geometric group theory and topology. Our main result is that every end-space can topologically be represented by a special order tree. Our main proof ingredient is a structure theorem that we introduce, which carves out the order-tree-like structure of any graph in such a way that there is a natural bijection between the ends of the graph and the limit-type down-closed chains of the order-tree.

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arXiv2023

The immersion-minimal infinitely edge-connected graph

We show that there is a unique immersion-minimal infinitely edge-connected graph: every such graph contains the halved Farey graph, which is itself infinitely edge-connected, as an immersion minor. By contrast, any minimal list of infinitely edge-connected graphs represented in all such graphs as topological minors must be uncountable.

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arXiv2023

Entanglements

Robertson and Seymour constructed for every graph $G$ a tree-decomposition that efficiently distinguishes all the tangles in $G$. While all previous constructions of these decompositions are either iterative in nature or not canonical, we give an explicit one-step construction that is canonical. The key ingredient is an axiomatisation of 'local properties' of tangles. Generalisations to locally finite graphs and matroids are also discussed.

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arXiv2023

The Lovász-Cherkassky theorem for locally finite graphs with ends

Lovász and Cherkassky discovered independently that, if $G$ is a finite graph and $T\subseteq V(G)$ such that the degree $d_G(v)$ is even for every vertex $v\in V(G)\setminus T$, then the maximum number of edge-disjoint paths which are internally disjoint from~$T$ and connect distinct vertices of $T$ is equal to $\frac{1}{2} \sum_{t\in T}λ_G(t, T\setminus \{t\})$ (where $λ_G(t, T\setminus \{t\})$ is the size of a smallest cut that separates $t$ and $T\setminus\{t\}$). From another perspective, this means that for every vertex $t\in T$, in any optimal path-system there are $λ_G(t, T\setminus \{t\})$ many paths between $t$ and~$T\setminus\{t\}$. We extend the theorem of Lovász and Cherkassky based on this reformulation to all locally-finite infinite graphs and their ends. In our generalisation, $T$ may contain not just vertices but ends as well, and paths are one-way (two-way) infinite when they establish a vertex-end (end-end) connection.

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arXiv2021

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.

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arXiv2021

A strengthening of Halin's grid theorem

We show that for every infinite collection $\mathcal{R}$ of disjoint equivalent rays in a graph $G$ there is a subdivision of the hexagonal half-grid in $G$ such that all its vertical rays belong to $\mathcal{R}$. This result strengthens Halin's grid theorem by giving control over which specific set of rays is used, while its proof is significantly shorter.

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arXiv2021

Approximating infinite graphs by normal trees

We show that every connected graph can be approximated by a normal tree, up to some arbitrarily small error phrased in terms of neighbourhoods around its ends. The existence of such approximate normal trees has consequences of both combinatorial and topological nature. On the combinatorial side, we show that a graph has a normal spanning tree as soon as it has normal spanning trees locally at each end; i.e., the only obstruction for a graph to having a normal spanning tree is an end for which none of its neighbourhoods has a normal spanning tree. On the topological side, we show that the end space $Ω(G)$, as well as the spaces $|G| = G \cup Ω(G)$ naturally associated with a graph $G$, are always paracompact. This gives unified and short proofs for a number of results by Diestel, Sprüssel and Polat, and answers an open question about metrizability of end spaces by Polat.

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arXiv2021

End-faithful spanning trees in graphs without normal spanning trees

Schmidt characterised the class of rayless graphs by an ordinal rank function, which makes it possible to prove statements about rayless graphs by transfinite induction. Halin asked whether Schmidt's rank function can be generalised to characterise other important classes of graphs. We answer Halin's question in the affirmative. Another largely open problem raised by Halin asks for a characterisation of the class of graphs with an end-faithful spanning tree. A well-studied subclass is formed by the graphs with a normal spanning tree. We determine a larger subclass, the class of normally traceable graphs, which consists of the connected graphs with a rayless tree-decomposition into normally spanned parts. Investigating the class of normally traceable graphs further we prove that, for every normally traceable graph, having a rayless spanning tree is equivalent to all its ends being dominated. Our proofs rely on a characterisation of the class of normally traceable graphs by an ordinal rank function that we provide.

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