On the Caccetta-Haggkvist conjecture with forbidden subgraphs

TL;DR

This paper advances the understanding of the Caccetta-Haggkvist conjecture for directed graphs by proving it for graphs missing specific small subgraphs, including certain cycles, thus narrowing the conditions under which the conjecture holds.

Contribution

The paper identifies three specific small subgraphs whose absence in a directed graph guarantees the conjecture's validity, extending previous results.

Findings

Proved the conjecture for graphs missing certain 4-vertex subgraphs.

Extended results to graphs missing cycles of length 3.

Provided a method to lift restrictions from induced subgraphs to general subgraphs.

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.