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Posted by: fhavet
on: February 26th, 2013

An antidirected tree is an orientation of a tree in which every vertex has either indegree 0 or outdergree 0.

Conjecture   Let $ D $ be a digraph. If $ |A(D)| > (k-2) |V(D)| $, then $ D $ contains every antidirected tree of order $ k $.

The value $ k-2 $ would be best possible, since the oriented tree consisting of a vertex dominating $ k-1 $ other vertices is not contained in any digraph in which every vertex has outdegree $ k-2 $. The condition on the trees be antidirected cannot be suppressed. In a bipartite digraph $ D $ with bipartition $ (A,B) $ such that all arcs are directed from $ A $ to $ B $, all the trees contained in $ D $ are antidirected.

This conjecture for symmetric digraphs is equivalent to the celebrated Erdös-Sos conjecture for undirected graphs. (see [E]).

Conjecture   Let $ G $ be a graph. If $ |E(G)| > \frac{1}{2} (k-2) |V(G)| $, then $ G $ contains every tree of order $ k $ .

Addario-Berry et al. Conjecture also implies Burr's conjecture (see Oriented trees in n-chromatic digraphs) for antidirected trees, since every digraph with chromatic number $ 2k-2 $ contains a colour-critical digraph has minimum degree at least $ 2k-3 $, and so whose number of vertices is at least $ \frac{2k-3}{2}|V(D)| $, which exceeds $ (k-2) |V(D)| $.

This conjecture has only been proved [AHL+] for antidirected trees of diameter at most $ 3 $.


*[AHL+] L. Addario-Berry, F. Havet, C. Linhares Sales, B. Reed, and S. Thomassé. Oriented trees in digraphs. Discrete Mathematics, 313(8):967-974, 2013.

[E] P. Erdös, Some problems in graph theory, Theory of Graphs and Its Applications, M. Fielder, Editor, Academic Press, New York, 1965, pp. 29--36.

* indicates original appearance(s) of problem.


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