Erdős-Posa property for long directed cycles

Importance: Medium ✭✭
Recomm. for undergrads: no
Posted by: fhavet
on: June 25th, 2013
Conjecture   Let $ \ell \geq 2 $ be an integer. For every integer $ n\geq 0 $, there exists an integer $ t_n=t_n(\ell) $ such that for every digraph $ D $, either $ D $ has a $ n $ pairwise-disjoint directed cycles of length at least $ \ell $, or there exists a set $ T $ of at most $ t_n $ vertices such that $ D-T $ has no directed cycles of length at least $ \ell $.

The case $ \ell=2 $ has been proved by Reed et al. [RRST], hence solving a conjecture of Gallai [G] and Younger [Y]. The case $ \ell=2 $ and $ n=2 $ has previously been solved by McCuaig [M], who proved that $ t_2(2)=3 $. Havet and Maia [HM] proved the case $ \ell=3 $.

The analogous statement for undirected graph has been proved by Birmelé, Bondy and Reed [BBR], hence generalizing Erdős-Posa [EP] result for $ \ell =3 $.


[BBR] E. Birmelé, J.A. Bondy, and B.A. Reed. The Erdos-Posa property for long circuits, Combinatorica, 27(2), 135–145, 2007.

[EP] P. Erdős and L. Pósa. On the independent circuits contained in a graph. Canad. J. Math., 17, 347--352, 1965.

[G] T. Gallai. Problem 6, in Theory of Graphs, Proc. Colloq. Tihany 1966 (New York), Academic Press, p.362, 1968.

*[HM] F. Havet and A. K. Maia. On disjoint directed cycles with prescribed minimum lengths. INRIA Research Report, RR-8286, 2013.

[M] W. McCuaig, Intercyclic digraphs. Graph Structure Theory, (Neil Robertson and Paul Seymour, eds.), AMS Contemporary Math., 147:203--245, 1993.

[RRST] B. Reed, N. Robertson, P.D. Seymour, and R. Thomas. Packing directed circuits. Combinatorica, 16(4):535--554, 1996.

[Y] D. H. Younger. Graphs with interlinked directed circuits. Proceedings of the Midwest Symposium on Circuit Theory, 2:XVI 2.1 - XVI 2.7, 1973.

* indicates original appearance(s) of problem.