Packing and covering of graphs
Final Report Abstract
Partition and decomposition problems as well as packing and covering problems have been an integral part of mathematics research ever since. This project contributed to these type of problems in graph theory and combinatorics. There are three main contributions. Firstly, Glock, Kim, Kühn, Osthus and myself solved the Oberwolfach problem, posed by Ringel in the 1960s, for all large instances. Secondly, joint work with Bruhn and Heinlein led to the result that edge-disjoint packings and coverings of graphs that contain a particular graph as a minor behave significantly different than their ‘vertex-disjoint’ analogue. In particular, we considered the case of graphs containing a certain tree as a minor and verified that even for these very simple graphs packing and covering relations that are well-known and well-behaved in the ‘vertex-disjoint’ setting are missing in the ‘edge-disjoint’ setting. Thirdly, Jenssen, Perkins and myself developed a novel method to investigate geometric packing problems. We gave the first improvement on the asymptotic behaviour in the dimension of the kissing numbers since the 1950s.
Publications
- A rainbow blow-up lemma
S. Glock and F. Joos
(See online at https://doi.org/10.1002/rsa.20907) - On kissing numbers and spherical codes in high dimensions. Advances in Mathematics, 335 (2018), 307–321
M. Jenssen, F. Joos, and W. Perkins
(See online at https://doi.org/10.1016/j.aim.2018.07.001) - Spanning trees in randomly perturbed graphs
F. Joos and J. Kim
(See online at https://doi.org/10.1002/rsa.20886) - On the hard sphere model and sphere packings in high dimensions. Forum Math. Sigma 7 (2019)
M. Jenssen, F. Joos, and W. Perkins
(See online at https://doi.org/10.1017/fms.2018.25)
