A Summary of Problems and Results related to the Caccetta-Haggkvist Conjecture

TL;DR

This paper surveys the current knowledge, open problems, and partial results concerning the Caccetta-Haggkvist conjecture and related graph theory conjectures, highlighting recent developments and connections to algebraic number theory.

Contribution

It provides a comprehensive summary of open problems, partial results, and connections to other conjectures related to the Caccetta-Haggkvist conjecture, based on a 2006 workshop.

Findings

Summary of partial results on the conjecture

Identification of open problems in the field

Connections to algebraic number theory and Cayley graphs

Abstract

This paper is an attempt to survey the current state of our knowledge on the Caccetta-Haggkvist conjecture and related questions. In January 2006 there was a workshop hosted by the American Institute of Mathematics in Palo Alto, on the Caccetta-Haggkvist conjecture, and this paper partly originated there, as a summary of the open problems and partial results presented at the workshop. This summary includes results and open problems related to Caccetta-Haggkvist, Seymour's Second Neighborhood Conjecture, the k/2 Conjecture (for nonedges), and connections with algebraic number theory through Cayley graphs, along with a number of other related topics.