let us suppose, without loss of generality, that T is an out-BFS-tree in D. The proof when T is an in-BFS-tree is analogous ,
, Since there is no arc from L i to L j for every j ? i + 2, the strong components of D 1 and D 2 are contained in the levels. Hence, by Lemma 29, ?(D 1 ) = max{ ?(D L i ) | i is odd} and ?(D 2 ) = max{ ?(D L i ) | i is even}. Moreover, Let D 1 and D 2 be the subdigraphs of D induced by the vertices of odd and even levels, respectively
, Let u be a vertex in D and T u an out-BFS-tree with root u. By Lemma 30, By Lemma, vol.29
Let T v be an in-BFS-tree in C rooted at v. By Lemma 30, there is a level L v of T v such that ?(D L v ) ? c. Now since mader ? (F ? a) = c, D L v contains a subdivision S of F ? a. With a slight abuse of notation ,
Oriented trees in digraphs, Discrete Mathematics, vol.313, issue.8, pp.967-974, 2013. ,
URL : https://hal.archives-ouvertes.fr/inria-00551133
The Erdos-Posa Property for Directed Graphs, 2016. ,
Disjoint directed cycles, Journal of Combinatorial Theory, Series B, vol.68, issue.2, pp.167-178, 1996. ,
Cycles in digraph -a survey, Journal of Graph Theory, vol.5, issue.1, pp.1-43, 1981. ,
Proof of a conjecture of Mader, Erdös and Hajnal on topological complete subgraphs, European Journal of Combinatorics, vol.19, issue.8, pp.883-887, 1998. ,
The Erdös-Sós conjecture for graphs of girth 5, Discrete Mathematics, vol.150, issue.1, pp.411-414, 1996. ,
Subtrees of directed graphs and hypergraphs, Proceedings of the Eleventh Southeastern Conference on Combinatorics, Graph Theory and Computing, vol.I, pp.227-239, 1980. ,
Antidirected subtrees of directed graphs, Canad. Math. Bull, vol.25, issue.1, pp.119-120, 1982. ,
Subdivisions of oriented cycles in digraphs with large chromatic number, 2016. ,
URL : https://hal.archives-ouvertes.fr/hal-01834779
Immersing complete digraphs, European Journal of Combinatorics, vol.33, issue.6, pp.1294-1302, 2012. ,
, abstrakten graphen vorhandene vollständige 4-graphen und ihre unterteilungen. Mathematische Nachrichten, vol.22, pp.61-85, 1960.
Some problems in graph theory, Theory of Graphs and Its Applications, pp.29-36, 1965. ,
On the representation of directed graphs as unions of orderings, Magyar Tud. Akad. Mat. Kutató Int. Közl, vol.9, pp.125-132, 1964. ,
Short proof of Menger's theorem, Discrete Mathematics, vol.219, issue.1-3, pp.295-296, 2000. ,
Tree embeddings, Journal of Graph Theory, vol.36, issue.3, pp.121-130, 2001. ,
The directed grid theorem, Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC '15, pp.655-664, 2015. ,
Topological cliques in graphs ii, Combinatorics, Probability and Computing, vol.5, pp.79-90, 1996. ,
Problem 2, Recent advances in graph theory, Proceedings of the Second Czechoslovak Symposium, 1974. ,
Homomorphieeigenschaften und mittlere kantendichte von graphen, Mathematische Annalen, vol.174, pp.265-268, 1967. ,
Degree and local connectivity in digraphs, Combinatorica, vol.5, issue.2, pp.161-165, 1985. ,
Existence of vertices of local connectivity k in digraphs of large outdegree, Combinatorica, vol.15, issue.4, pp.533-539, 1995. ,
Zur allgemeinen kurventheorie, Fundamenta Mathematicae, vol.10, issue.1, pp.96-115, 1927. ,
, Packing directed circuits. Combinatorica, vol.16, issue.4, pp.535-554, 1996.
The Erdös-Sós conjecture for graphs without c 4, Journal of Combinatorial Theory, Series B, vol.70, issue.2, pp.367-372, 1997. ,
Characterization of even directed graphs, Journal of Combinatorial Theory, Series B, vol.42, issue.1, pp.36-45, 1987. ,
Some homeomorphism properties of graphs, Mathematische Nachrichten, vol.64, issue.1, pp.119-133, 1974. ,
Sign-nonsingular matrices and even cycles in directed graphs, Linear Algebra and its Applications, vol.75, pp.27-41, 1986. ,
Configurations in graphs of large minimum degree, connectivity, or chromatic number, Annals of the New York Academy of Sciences, vol.555, issue.1, pp.402-412, 1989. ,
K5-subdivisions in graphs, Combinatorics, Probability and Computing, vol.5, pp.179-189, 1996. ,