[Paper Review] A Summary of Problems and Results related to the Caccetta-Haggkvist Conjecture

Blair D. Sullivan|ArXiv.org|May 24, 2006

Advanced Combinatorial Mathematics24 references20 citations

TL;DR

This paper provides a comprehensive survey of the Caccetta-H"{a}ggkvist conjecture and related problems in directed graph theory, synthesizing partial results, approximate bounds, and open conjectures. It consolidates progress on minimum out-degree conditions forcing short directed cycles, including exact results for small r, approximate bounds with small additive constants, and connections to Seymour's second neighborhood conjecture and r-regular digraphs.

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.

Motivation & Objective

• To summarize the state of knowledge on the Caccetta-H"{a}ggkvist conjecture as of January 2006, following a workshop at the American Institute of Mathematics.
• To compile and systematize partial results, approximate bounds, and related conjectures connected to the conjecture.
• To clarify connections between the Caccetta-H"{a}ggkvist conjecture and other central problems in directed graph theory, such as Seymour's second neighborhood conjecture and the Behzad-Chartrand-Wall conjecture.
• To present open problems and conjectures in a structured way to guide future research in extremal digraph theory.
• To provide a unified reference for researchers on the current status of the conjecture and its implications across graph classes and degree conditions.

Proposed method

• Surveying and organizing known results on the Caccetta-H"{a}ggkvist conjecture, including proofs for small r (r=2,3,4,5) and asymptotic bounds for r ≤ √(n/2).
• Analyzing approximate results that bound the girth by n/r + c for small constants c (e.g., c=73 by Shen), using combinatorial and probabilistic techniques.
• Examining special cases such as r = n/3, where the goal is to minimize the constant c such that δ⁺ ≥ cn forces a directed cycle of length at most 3.
• Investigating connections to Seymour's second neighborhood conjecture, including probabilistic and algebraic methods to bound the size of second neighborhoods.
• Presenting structural conjectures such as Lichiardopol's conjecture on feedback arc sets and Thomassé's conjecture on path lengths in relation to girth.
• Introducing and analyzing the class of 3/4-digraphs and majority digraphs as potential candidates for extremal examples, with conjectures on their structural properties.

Experimental results

Research questions

• RQ1 What is the minimal length of a directed cycle in a simple n-vertex digraph with minimum out-degree r, and how does it relate to the bound ⌈n/r⌉?
• RQ2 Can the Caccetta-H"{a}ggkvist conjecture be proven for all r, or are there only finitely many counterexamples for each r?
• RQ3 What is the best possible additive constant c such that δ⁺ ≥ r implies a directed cycle of length at most n/r + c?
• RQ4 Does every digraph with minimum out-degree at least n/3 contain a directed triangle, and what is the minimal such constant?
• RQ5 Do 3/4-digraphs—defined via majority of permutations—satisfy the Caccetta-H"{a}ggkvist conjecture, and what structural properties do they possess?

Key findings

• The Caccetta-H"{a}ggkvist conjecture holds for r=2, r=3, r=4, and r=5, with proofs by Caccetta and H"{a}ggkvist, Hamidoune, and Hoañg and Reed, respectively.
• Shen proved that for n ≥ 2r² − 3r + 1, any digraph with minimum out-degree r has a cycle of length at most ⌈n/r⌉, establishing a finite exception set for each r.
• For the r = n/3 case, the best known upper bound on the constant c such that δ⁺ ≥ cn forces a directed cycle of length at most 3 is c ≤ 3 − √7 ≈ 0.3542.
• Shen's result on approximate girth gives g ≤ 3⌈n/r ln((2+√7)/3)⌉ ≈ 1.312n/r, improving earlier bounds.
• Seymour's second neighborhood conjecture is known to hold for tournaments (Dean's conjecture), digraphs with out-degree ≤ 6, and for a fraction γ ≈ 0.657 of the first neighborhood size.
• Lichiardopol's conjecture implies the Caccetta-H"{a}ggkvist conjecture and posits that every digraph has a minimal feedback arc set containing a path of length δ⁺.