Graph Theory » Directed Graphs

Caccetta-Häggkvist Conjecture ★★★★

Author(s): Caccetta; Häggkvist

Conjecture   Every simple digraph of order $ n $ with minimum outdegree at least $ r $ has a cycle with length at most $ \lceil n/r\rceil $

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Posted by fhavet
updated February 28th, 2013
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Graph Theory » Directed Graphs

Directed path of length twice the minimum outdegree ★★★

Author(s): Thomassé

Conjecture   Every oriented graph with minimum outdegree $ k $ contains a directed path of length $ 2k $.

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Posted by fhavet
updated February 28th, 2013
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Graph Theory » Directed Graphs

Antidirected trees in digraphs ★★

Author(s): Addario-Berry; Havet; Linhares Sales; Reed; Thomassé

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 $.

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Posted by fhavet
updated February 26th, 2013
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