Dr. Jan Kurkofka 
Upcoming events

EMail: j.lastname (at) bham.ac.uk 
Selected Publications
 Canonical graph decompositions via coverings (with Reinhard Diestel, Raphael W. Jacobs and Paul Knappe), submitted (2022). (arXiv)
 Canonical decompositions of 3connected graphs (with Johannes Carmesin), accepted to FOCS 2023. (arXiv)
 Every infinitely edgeconnected graph contains the Farey graph or $T_{\aleph_0}\!\ast t$ as a minor, Mathematische Annalen 382 (2022), 18811900. ( Journal  arXiv )
Teaching

Selected talks

Research interests
My research interests include Graph Minor Theory, Connectivity Theory, Topological Graph Theory, Infinite Graph Theory, and intradisciplinary applications thereof. Here are some ongoing and recent projects:
Explicit structure theorems for high connectivity
Decomposing graphs into highly connected regions is a foundational method in Graph Minor Theory, and Connectivity Theory more broadly. Explicit solutions to the following problem for any k would be tremendously useful for computing excluded minors: Decompose every kconnected graph along kseparators into smaller pieces that are (k + 1)connected or 'basic'. The first step k = 1 is given by the blockcutvertex decomposition. For k = 2, the solution is a classical theorem of Tutte from 1966, which builds on earlier works of Whitney from the 1930s. It was long believed that Tutte's Theorem cannot be fully extended to higher connectivity. I recently proposed a new perspective on graph connectivity that allowed me to prove a TutteTheorem for k = 3 with Carmesin. Can this be extended to larger k ?


Localglobal graph theory
The question to what extent graph invariants  the chromatic number, say, or connectivity  are of local or global character,
and how their local and global aspects interact, drives much of the research in graph theory both structural and extremal. This project offers such studies a possible formal basis.
In our main paper we combine coverings, as known from Topology, with tangletree decompositions as known from Graph Minor Theory, to canonically decompose finite graphs and groups into their highly connected local parts. The global structure of the graph or group, as determined by the relative position of these parts,
is then described by a coarser model. I am currently working on applications of our result and methodoloy in Computer Science and Combinatorial Group Theory.


Farey graph
The Farey graph, shown on the left and surveyed on 300 pages in Hatcher's new book
(PDF),
plays a role in a number of mathematical fields ranging from group theory and number theory to geometry and dynamics.
Curiously, graph theory has not been among these until very recently, when I showed that the Farey graph plays a central role in graph theory too:
it is one of two infinitely edgeconnected graphs that must occur as a minor in every infinitely edgeconnected graph.
Previously it was not known that there was any set of graphs determining infinite edgeconnectivity by forming a minorminimal list in this way, let alone a finite set.
This resulted marked the starting point of a project in which I addressed exciting followup questions.


The whirl graph on the left answers three questions about the Farey graph at once.
For instance, the whirl graph is infinitely edgeconnected and contains the Farey graph as a minor with branch sets of size two, but it does not contain a subdivision of the Farey graph.
In fact, the whirl graph contains no subdivision of any naively constructed infinitely edgeconnected graph, because it has the following property:
For any two vertices u,v and any positive integer k, the whirl graph contains k edgedisjoint ordercompatible u–v paths but not infinitely many.

Publications and preprints
Explicit structure theorems for high connectivity
 Canonical decompositions of 3connected graphs (with Johannes Carmesin), accepted to FOCS 2023. (arXiv)
 Entanglements (with Johannes Carmesin), Journal of Combinatorial Theory, Series B 164 (2024), 1728. ( Journal  arXiv )
 Characterising 4tangles through a connectivity property (with Johannes Carmesin), submitted (2023). (arXiv)
Localglobal graph theory
 Canonical graph decompositions via coverings (with Reinhard Diestel, Raphael W. Jacobs and Paul Knappe), submitted (2022). (arXiv)
Combinatorics in 3D
 On the edgechromatic number of 2complexes (with Emily Nevinson), submitted (2023). (arXiv)
Farey graph
 Every infinitely edgeconnected graph contains the Farey graph or $T_{\aleph_0}\!\ast t$ as a minor, Mathematische Annalen 382 (2022), 18811900. ( Journal  arXiv )
 The immersionminimal infinitely edgeconnected graph (with Paul Knappe), Journal of Combinatorial Theory, Series B 164 (2024), 492516. ( Journal  arXiv )
 The Farey graph is uniquely determined by its connectivity, Journal of Combinatorial Theory, Series B 151 (2021), 223234. ( Journal  arXiv )
 Ubiquity and the Farey graph, European Journal of Combinatorics 95 (2021), 103326. ( Journal  arXiv )
End spaces
 A representation theorem for end spaces of infinite graphs (with Max Pitz), submitted (2021). (arXiv)
 Halin's end degree conjecture (with Stefan Geschke, Ruben Melcher and Max Pitz), Israel Journal of Mathematics (2022). ( Journal  arXiv )
 Approximating infinite graphs by normal trees (with Ruben Melcher and Max Pitz), Journal of Combinatorial Theory, Series B 148 (2021), 173183. ( Journal  arXiv )
 Countably determined ends and graphs (with Ruben Melcher), Journal of Combinatorial Theory, Series B 156 (2022), 3156. ( Journal  arXiv )
 Endfaithful spanning trees in graphs without normal spanning trees (with Carl Bürger), Journal of Graph Theory (2022). ( Journal  arXiv )
 A strengthening of Halin's grid theorem (with Ruben Melcher and Max Pitz), Mathematika 68(4) (2022). ( Journal  arXiv )
Starcomb series
 Duality theorems for stars and combs I: Arbitrary stars and combs (with Carl Bürger), Journal of Graph Theory 99(4) (2022), 525554. ( Journal  arXiv )
 Duality theorems for stars and combs II: Dominating stars and dominated combs (with Carl Bürger), Journal of Graph Theory 99(4) (2022), 555572. ( Journal  arXiv )
 Duality theorems for stars and combs III: Undominated combs (with Carl Bürger), Journal of Graph Theory 100(1) (2022), 127139. ( Journal  arXiv)
 Duality theorems for stars and combs IV: Undominating stars (with Carl Bürger), Journal of Graph Theory 100(1) (2022), 140162. ( Journal  arXiv)
Ends and tangles
 Tangles and the StoneČech compactification of infinite graphs (with Max Pitz), Journal of Combinatorial Theory, Series B 146 (2021), 3460. ( Journal  arXiv )
 Ends, tangles and critical vertex sets (with Max Pitz), Mathematische Nachrichten 292(9) (2019), 20722091. ( Journal  arXiv )
 A treeoftangles theorem for infinite tangles (with AnnKathrin Elm), Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 92 (2022), 139178. ( Journal  arXiv )
General connectivity
 The LovászCherkassky theorem for locally finite graphs with ends (with Raphael W. Jacobs, Attila Joó, Paul Knappe and Ruben Melcher), Discrete Mathematics 346(12) (2023), 113586. ( Journal  arXiv )
 Edgeconnectivity and treestructure in finite and infinite graphs (with Christian Elbracht and Maximilian Teegen), 2021. (arXiv)
Theses