On the Caccetta–Häggkvist conjecture with forbidden subgraphs

The Caccetta–Häggkvist conjecture developed in 1978 asserts that every oriented graph on $n$ vertices without oriented cycles of length $\leqslant \ell$ must contain a vertex of outdegree at most $\frac{n-1}{\ell}$. It has a rather elaborate set of (conjectured) extremal configurations. In this paper, we consider the case $\ell=3$ that received quite a significant attention in the literature. We identify three oriented graphs on four vertices each that are missing as an induced subgraph in all known extremal examples and prove the Caccetta–Häggkvist conjecture for oriented graphs missing as induced subgraphs any of these oriented graphs, along with $C_3$. Using a standard method, we can also lift the restriction of being induced, though this makes graphs in our list slightly more complicated.