We construct families of circles in the plane such that their tangency graphs have arbitrarily large girth and chromatic number. This provides a strong negative answer to Ringel’s circle problem (1959). The proof relies on a (multidimensional) version of Gallai’s theorem with polynomial constraints, which we derive from the Hales-Jewett theorem and which may be of independent interest.
@InProceedings{davies_et_al:LIPIcs.SoCG.2022.33, author = {Davies, James and Keller, Chaya and Kleist, Linda and Smorodinsky, Shakhar and Walczak, Bartosz}, title = {{A Solution to Ringel’s Circle Problem}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {33:1--33:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.33}, URN = {urn:nbn:de:0030-drops-160413}, doi = {10.4230/LIPIcs.SoCG.2022.33}, annote = {Keywords: circle arrangement, chromatic number, Gallai’s theorem, polynomial method} }
Feedback for Dagstuhl Publishing