On Iterated Image Size for Point-Symmetric Relations

Let Γ = (V,E) be a point-symmetric reflexive relation and let v ∈ V such that |Γ(v)| is finite (and hence |Γ(x)| is finite for all x , by the transitive action of the group of automorphisms). Let j ∈ N be an integer such that Γ(v)∩Γ(v) = {v}. Our main result states that |Γ(v)| ≥ |Γ(v)|+ |Γ(v)| − 1. As an application we have |Γ(v)| ≥ 1+(|Γ(v)|−1)j. The last result confirms a recent conjecture of Seymour in the case of vertex-symmetric graphs. Also it gives a short proof for the validity of the Caccetta-Häggkvist conjecture for vertex-symmetric graphs and generalizes an additive result of Shepherdson.