• Corpus ID: 2833200

# The Caccetta-Haggkvist conjecture and additive number theory

@article{Nathanson2006TheCC,
title={The Caccetta-Haggkvist conjecture and additive number theory},
author={Melvyn B. Nathanson},
journal={arXiv: Combinatorics},
year={2006}
}
• M. Nathanson
• Published 20 March 2006
• Mathematics
• arXiv: Combinatorics
The Caccetta-Haggkvist conjecture states that if G is a finite directed graph with at least n/k edges going out of each vertex, then G contains a directed cycle of length at most k. Hamidoune used methods and results from additive number theory to prove the conjecture for Cayley graphs and for vertex-transitive graphs. This expository paper contains a survey of results on the Caccetta-Haggkvist conjecture, and complete proofs of the conjecture in the case of Cayley and vertex-transitive graphs.
A Cauchy–Davenport Type Result for Arbitrary Regular Graphs
There is a universal constant ∈ > 0 such that, if is a connected, regular graph on n vertices, then either every pair of vertices can be connected by a path of length at most three, or the number of pairs of such vertices is at least 1 + ∈ times thenumber of edges in .
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Let G be a finite simple directed graph on n vertices. Say G is m-free if it has no directed cycles of length at most m. In 1978, Caccetta and Haggkvist [3] conjectured that if G has minimum
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This paper gives a short proof for the validity of the Caccetta–Häggkvist conjecture for vertex-symmetric graphs and generalizes an additive result of Shepherdson.

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