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Subdivision of a transitive tournament in digraphs with large outdegree.

Importance: Medium ✭✭
Author(s): Mader, W.
Subject: Graph Theory
» Directed Graphs
Keywords:
Recomm. for undergrads: no
Posted by: fhavet
on: March 4th, 2013
Conjecture   For all $ k $ there is an integer $ f(k) $ such that every digraph of minimum outdegree at least $ f(k) $ contains a subdivision of a transitive tournament of order $ k $.

A fundamental result of Mader [M1] states that for every integer $ k $ there is a smallest $ g(k) $ so that every graph of average degree at least $ g(k) $ contains a subdivision of a complete graph on $ k $ vertices. Bollobás and Thomason [BT] as well as Komlós and Szemerédi [KS] showed that $ g $ is quadratic in $ k $.

The above conjecture is a digraph analogue of this result. However one cannot replace the minimum outdegree in this conjecture by the average degree as in Mader's analogue for graphs: consider the complete bipartite graph $ K_{n,n} $ and orient all edges from the first to the second class. The resulting digraph has average degree $ n $ but not even a transitive tournament on 3 vertices.

One might be tempted to conjecture that large minimum outdegree would even force the existence of a subdivision of a large complete digraph. However, for all $ n $ Thomassen [T] constructed a digraph on $ n $ vertices whose minimum outdegree is at least $ \frac{1}{2} \log_2 n $ but which does not contain an even directed cycle (and thus no complete digraph on 3 vertices). A simpler construction was found by DeVos et al. [DMMS].

It is easy to see that $ f(1)=0 $ and $ f(2)=1 $. Mader [M3] showed that $ f(4) = 3 $. Even the existence of $ f(5) $ is not known.

Bibliography

[BT] B. Bollobás and A. Thomason, Proof of a conjecture of Mader, Erdös and Hajnal on topological complete subgraphs, European Journal of Combinatorics 19 (1998), 883–887.

[DMMS] M. DeVos, J. McDonald, B. Mohar, and D. Scheide, Immersing complete digraphs, European Journal of Combinatorics, 33 (2012), no 6, 1294-1302.

[KS] J. Komlós and E. Szemerédi, Topological Cliques in Graphs II, Combinatorics, Probability and Computing 5 (1996), 70–90.

[M1] W. Mader, Homomorphieeigenschaften und mittlere Kantendichte von Graphen, Math. Annalen 174 (1967), 265–268.

* [M2] W. Mader, Degree and Local Connectivity in Digraphs, Combinatorica 5 (1985), 161–165.

[M3] W. Mader, On Topological Tournaments of order 4 in Digraphs of Outdegree 3, Journal of Graph Theory 21 (1996), 371–376.

[T] C. Thomassen, Even Cycles in Directed Graphs, European Journal of Combinatorics 6 (1985), 85–89.


* indicates original appearance(s) of problem.
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