Florian Lehner's Webpage

About me

Florian Lehner

I am a lecturer at the Department of Mathematics at the University of Auckland.

My research interests cover a wide variety of topics in graph theory and combinatorics. Much of my work concerns problems on the interface of group theory and graph theory, but I also enjoy working on topics with a geometric or probabilistic flavour.

Recent research

Asymmetric graph colouring

some distinguishing colourings How hard is it, to get rid of the symmetries of a discrete structure? More precisely, how many colours do we need to colour the vertices of a graph in a way that no automorphism except the identity preserves the colouring?

The least number of colours required is called the asymmetric colouring number or distinguishing number of the graph. I have studied this graph invariant in various settings, including random colourings, edge colourings, and most recently colourings of graphs with bounded maximum degree.

Combinatorial game theory

cops-and-robber game The study of pursuit-evasion games on graphs is a popular topic in combinatorial game theory. These are games where one player's objective is to capture the other player, whereas the other player attempts to avoid being captured. Notable examples include Cops-and-Robber and Conway's Angel-Devil game.

My contributions mostly concern the so-called cop number, that is, the least number of pursuers needed to guarantee capture in the Cops-and-Robber game, but I am always interested in learning about other aspects of the game as well as other combinatorial games.

Tree structure in infinite graphs

amalgamated free product structure Stallings' splitting theorem is one of the central results in geometric group theory. It essentially states that a finitely generated group satisfies one of two alternatives. It either has one well-connected infinite part (that is, it is one-ended), or it exhibits a large-scale tree structure (either an amalgamated free product, or a HNN extension).

Hamann, Miraftab, Rühmann, and I recently proved an analogous result for infinite graphs; I am currently working on applications as well as further extensions of this result.

Resume

You can also download a CV in PDF format.

Employment

Lecturer Department of Mathematics, University of Auckland since 2022
Project Assistant Department of Discrete Mathematics, Graz University of Technology 2020 – 2022
Erwin Schrödinger Research Fellow Department of Mathematics, University of Warwick Department of Discrete Mathematics, Graz University of Technology 2017 – 2020
University Assistant Department of Mathematics, University of Hamburg 2015 – 2017
University Assistant Institute of Geometry, Graz University of Technology 2011 – 2015
Teaching Assistant Mathematics Institutes and Institute for Software Technology, Graz University of Technology 2007 – 2010

Education

PhD: Mathematics and Scientific Computing Graz University of Technology Graduation with highest distinctions "Sub auspiciis Praesidentis" Thesis: Symmetry breaking in graphs and groups. Adviser: J. Wallner 2011 – 2014
Master Programme: Mathematical Computer Science Graz University of Technology Thesis: The line graph of every locally finite 6-edge-connected graph with finitely many ends is hamiltonian. Adviser: W. Woess 2008 – 2011
Bachelor Programme: Technical Mathematics Graz University of Technology 2005 – 2008

Publications

Articles

[46] Sparse cycle bases for graphs with bounded genus. with B. Miraftab 2024, preprint. pdf
[45] Locally finite graphs and their localization numbers. with A. Bonato, T. G. Marbach, and JD Nir 2024, preprint. pdf
[44] Self-avoiding walk is ballistic on graphs with more than one end. with C. Lindorfer and C. Panagiotis 2024, preprint. pdf
[43] Folding polyominoes into cubes. with O. Aichholzer and C. Lindorfer 2024, preprint. pdf
[42] Catching a robber on a random k-uniform hypergraph. with J. Erde, M. Kang, B. Mohar, and D. Schmid Canadian Journal of Mathematics, 2024, to appear. pdf
[41] Chromatic number is not tournament-local. with A. Girão, K. Hendrey, F. Illingworth, L. Michel, M. Savery, and R. Steiner Journal of Combinatorial Theory, Series B, 2024, to appear. pdf
[40] Asymmetrizing infinite trees. with W. Imrich, R. Kalinowski, M. Pilśniak, and M. Stawiski 2023, preprint. pdf
[39] Tipsy cop and tipsy robber: collisions of biased random walks on graphs. with P. E. Harris and E. Insko Theoretical Computer Science, 2024, to appear. pdf
[38] Asymmetric colouring of locally compact permutation groups. Bulletin of the London Mathematical Society, 55(4), pp. 1874 – 1879, 2023. doi pdf
[37] A note on classes of subgraphs of locally finite graphs. Journal of Combinatorial Theory, Series B, 161, pp. 52 – 62, 2023. doi pdf
[36] Universal planar graphs for the topological minor relation. Combinatorica, 44, pp. 209 – 230, 2023. doi pdf
[35] Self-avoiding walks and multiple context-free languages. with C. Lindorfer Combinatorial Theory, 3 (1), 2023. doi pdf
[34] Hamiltonian decompositions of 4-regular Cayley graphs of infinite abelian groups. with J. Erde Journal of Graph Theory, 101 (3), pp. 559 – 571, 2022. doi pdf
[33] On fixity of arc-transitive graphs. with P. Potočnik and P. Spiga Science China Mathematics, 64, pp. 2603 – 2610, 2021. doi pdf
[32] Counterexamples to "A Conjecture on Induced Subgraphs of Cayley Graphs". with G. Verret Ars Mathematica Contemporanea, 19 (1), pp. 77 – 82, 2020. doi pdf
[31] Comparing consecutive letter counts in multiple context-free languages. with C. Lindorfer Theoretical Computer Science, 868, pp. 1 – 5, 2021. doi pdf
[30] On asymmetric colourings of graphs with bounded degrees and infinite motion. with M. Pilśniak and M. Stawiski Ars Mathematica Contemporanea, 2024, to appear. pdf
[29] A bound for the distinguishing index of regular graphs. with M. Pilśniak and M. Stawiski European Journal of Combinatorics, 89, 2020. doi pdf
[28] Bounding the cop number of a graph by its genus. with N. Bowler, J. Erde, and M. Pitz SIAM Journal on Discrete Mathematics, 35 (4), pp. 2459 – 2489, 2021. doi pdf
[27] On the cop number of toroidal graphs. Journal of Combinatorial Theory, Series B, 151, pp. 250 – 262, 2021. doi pdf
[26] A Stallings' type theorem for quasi-transitive graphs. with M. Hamann, B. Miraftab, and T. Rühmann Journal of Combinatorial Theory, Series B, 157, pp. 40 – 69, 2022. doi pdf
[25] Invariant spanning double rays in amenable groups. with A. Georgakopoulos Discrete Mathematics, 344(2), 2021. doi pdf
[24] Distinguishing infinite graphs with bounded degrees. with M. Pilśniak and M. Stawiski Journal of Graph Theory, 101(1), pp. 52 – 65, 2022. doi pdf
[23] Distinguishing numbers of finite 4-valent vertex-transitive graphs. with G. Verret Ars Mathematica Contemporanea, 19 (2), pp. 173 – 187, 2020. doi pdf
[22] On symmetries of edge and vertex colourings of graphs. with S. M. Smith Discrete Mathematics, 343 (9), 2020. doi pdf
[21] Distinguishing density and the Distinct Spheres Condition. with W. Imrich and S. M. Smith European Journal of Combinatorics, 89, 2020. doi pdf
[20] Hamilton decompositions of one-ended Cayley graphs. with J. Erde and M. Pitz Journal of Combinatorial Theory, Series B, 140, pp. 171 – 191, 2020. doi pdf
[19] Trees with distinguishing index equal distinguishing number plus one. with S. Alikhani, S. Klavžar, and S. Soltani Discussiones Mathematicae Graph Theory, 40 (3), pp. 875 – 884, 2020. doi pdf
[18] Firefighting on trees and Cayley graphs. Australasian Journal of Combinatorics, 75 (1), pp. 66 – 72, 2019. jrnl pdf
[17] On tree-decompositions of one-ended graphs. with J. Carmesin and R. G. Möller Mathematische Nachrichten, 292 (3), pp. 524 – 539, 2018. doi pdf
[16] A counterexample to Montgomery's conjecture on dynamic colourings of regular graphs. with N. Bowler, J. Erde, M. Merker, M. Pitz, and K. Stavropoulos Discrete Applied Mathematics, 229, pp. 151 – 153, 2017. doi pdf
[15] Non-reconstructible locally finite graphs. with N. Bowler, J. Erde, P. Heinig, and M. Pitz Journal of Combinatorial Theory, Series B, 133, pp. 122 – 152, 2018. doi pdf
[14] A counterexample to the reconstruction conjecture for locally finite trees. with N. Bowler, J. Erde, P. Heinig, and M. Pitz Bulletin of the London Mathematical Society, 49 (4), pp. 630 – 648, 2017. doi pdf
[13] Breaking graph symmetries by edge colourings. Journal of Combinatorial Theory, Series B, 127, pp. 205 – 214, 2017. doi pdf
[12] Fast factorization of Cartesian products of (directed) hypergraphs. with M. Hellmuth Theoretical Computer Science, 615, pp. 1 – 11, 2016. doi pdf
[11] Maximising the number of independent sets in connected graphs. with S. Wagner Graphs and Combinatorics, 33 (5), pp. 1103 – 1118, 2017. doi pdf
[10] Local finiteness, distinguishing numbers and Tucker's conjecture. with R. G. Möller Electronic Journal of Combinatorics, 22 (4), 2015, research paper #P4.19. doi pdf
[9] Pursuit evasion on infinite graphs. Theoretical Computer Science, 655 (Part A), pp. 30 – 40, 2016. doi pdf
[8] The Cartesian product of graphs with loops. with T. Boiko, J. Cuno, W. Imrich, and C. E. van de Woestijne Ars Mathematica Contemporanea, 11 (1), pp. 1 – 9, 2016. doi pdf
[7] Clique trees of infinite locally finite chordal graphs. with C. Temmel Electronic Journal of Combinatorics, 25 (2), 2018, research paper #P2.9. doi pdf
[6] Extending cycles locally to Hamilton cycles. with M. Hamann and J. Pott Electronic Journal of Combinatorics, 23 (1), 2016, research paper #P1.49. doi pdf
[5] Random colorings and automorphism breaking in locally finite graphs Combinatorics, Probability and Computing, 22 (6), pp. 885 – 909, 2013. doi pdf
[4] Distinguishing graphs with intermediate growth. Combinatorica, 36 (3), pp. 333 – 347, 2016. doi pdf
[3] Endomorphism breaking in graphs. with W. Imrich, R. Kalinowski, and M. Pilśniak Electronic Journal of Combinatorics, 21 (1), 2014, research paper #P1.16. doi pdf
[2] Distinguishing graphs with infinite motion and nonlinear growth. with J. Cuno and W. Imrich Ars Mathematica Contemporanea, 7, pp. 201 – 213, 2014. doi pdf
[1] On spanning tree packings of highly edge connected graphs. Journal of Combinatorial Theory, Series B, 105, pp. 93 – 126, 2014. doi pdf

Theses

[2] Symmetry breaking in graphs and groups Ph.D. Thesis, TU Graz, 2014. pdf
[1] The line graph of every locally finite 6-edge-connected graph with finitely many ends is hamiltonian. Master thesis, TU Graz, 2011. pdf