The circumference of a graph G is the length of a longest cycle in G, or \(+\infty \) if G has no cycle. Birmelé (J Graph Theory 43(1):24–25, 2003) showed that the treewidth of a graph G is at most its circumference minus 1. We strengthen this result for 2-connected graphs as follows: If G is 2-connected, then its treedepth is at most its circumference. The bound is best possible and improves on an earlier quadratic upper bound due to Marshall and Wood (J Graph Theory 79(3):222–232, 2015).
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This research was started at the Structural Graph Theory workshop in Gułtowy (Poland) in June 2022 organized by Andrzej Grzesik, Marcin Pilipczuk, and Marcin Witkowski. We thank the organizers and the other workshop participants for creating a productive working atmosphere. We thank Michał Pilipczuk for pointing out to us the algorithmic application mentioned in the introduction. We are grateful to the two anonymous referees for their helpful comments.
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G. Joret is supported by a CDR grant from the Belgian National Fund for Scientific Research (FNRS), a PDR grant from FNRS, and by the Wallonia Brussels International (WBI) agency. This work is a part of Project BOBR (K. Majewski) that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 948057). P. Micek, M. T. Seweryn and M. Briański are supported by the National Science Center of Poland under Grant UMO-2018/31/G/ST1/03718 within the BEETHOVEN program.
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Briański, M., Joret, G., Majewski, K. et al. Treedepth vs Circumference. Combinatorica 43, 659–664 (2023). https://doi.org/10.1007/s00493-023-00028-5