User:JoshuaZ/Caccetta-Häggkvist Conjecture

The Caccetta-Häggkvist Conjecture is a conjecture about directed graphs. The conjecture states that if one has an n-vertex graph with outdegree at least $$r$$ at every vertex, then the graph has a cycle with length at most the ceiling of $$\frac{n}{r}$$. The conjecture is trivial for $$r=1$$ and has been proven for various values of $$r$$. Caccetta and Haggkvist proved the conjecture for $$r=2$$ and subsequent work proved the result for $$r \leq 5$$ as well as $$r \leq \left(\frac{n}{2}\right)^{1/2}$$. This last result is a corollary of a more technical result of Shen.

http://www.aimath.org/pastworkshops/caccettarep.pdf http://www.math.uiuc.edu/~west/openp/cacchagg.html http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B75GV-4PCHPN5-35&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=3475dd59e6136f411c2b4a5851276f20