On the Caccetta-Häggkvist Conjecture with Forbidden Subgraphs

The Caccetta-Haggkvist conjecture made in 1978 asserts that ev- ery 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 congurations. In this paper we consider the case l = 3 that received quite a signicant attention in the literature. We identify three orgraphs on four vertices 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 C3. Using a standard trick, we can also lift the restriction of being induced, although this makes graphs in our list slightly more complicated.

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