The Caccetta-häggkvist Conjecture and Additive Number Theory

The Caccetta-Häggkvist 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-Häggkvist conjecture, and complete proofs of the conjecture in the case of Cayley and vertex-transitive graphs. 1. Many edges imply short cycles A finite directed graph G = (V,E) consists of a finite set V = V (G) of vertices and a finite set E = E(G) of edges, where an edge e = (v, v) is an ordered pair of vertices. If e = (v, v) is an edge, then the vertex v is called the tail of e, and v is called the head of e. The outdegree of a vertex v ∈ V , denoted outdegG(v), is the number of edges e ∈ E of the form (v, v), that is the number of edges with tail v. The indegree of a vertex v ∈ V , denoted indegG(v′), is the number of edges e ∈ E of the form (v, v), that is the number of edges with head v. Let v and v be distinct vertices of the finite directed graph G. A directed path of length l in G from vertex v to vertex v is a sequence of l edges (v0, v1), (v1, v2), . . . , (vl−1, vl) such that v = v0 and v ′ = vl. A directed cycle of length l in G is a sequence of l edges (v0, v1), (v1, v2), . . . , (vl−1, vl) such that v0 = vl. A loop is a cycle of length 1, that is, an edge of the form (v, v). A cycle of length 2 is called a digon, and consists of two edges of the form (v0, v1) and (v1, v0), where v0 6= v1. A directed triangle is a cycle of length 3 of the form (v0, v1), (v1, v2), (v2, v0), where the vertices v0, v1, v2 are distinct. It is reasonable to expect that a finite directed graph with many edges should have many cycles, and, in particular, should have short cycles. A quantitative expression of this intuition is the following: 2000 Mathematics Subject Classification. 05C20,05C25,11B13,11P99.