On the Caccetta-Haggkvist conjecture with forbidden subgraphs
arXiv:1107.2247
Abstract
The Caccetta-Haggkvist conjecture made in 1978 asserts that every orgraph on n vertices without oriented cycles of length <= l must contain a vertex of outdegree at most (n-1)/l. It has a rather elaborate set of (conjectured) extremal configurations. In this paper we consider the case l=3 that received quite a significant attention in the literature. We identify three orgraphs on four vertices each that are missing as an induced subgraph in all known extremal examples and prove the Caccetta-Haggkvist conjecture for orgraphs missing as induced subgraphs any of these orgraphs, along with cycles of length 3. Using a standard trick, we can also lift the restriction of being induced, although this makes graphs in our list slightly more complicated.
Referees' comments (the paper has been accepted to "Journal of Graph Theory") have been taken into account. Most notably, an annoying misprint in the proof of Claim 5.6 (K_{1,2} should everywhere read K_{2,1}) has been fixed