On iterated image size for point-symmetric relations
Abstract
Description
Let $Γ=(V,E)$ be a point-symmetric reflexive relation and let $v\in V$ 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 $j\in \N$ be an integer such that $Γ^j(v)\cap Γ^{-}(v)=\{v\}$. Our main result states that
$$ |Γ^{j} (v)|\ge | Γ^{j-1} (v)| + |Γ(v)|-1.$$
As an application we have $ |Γ^{j} (v)| \ge 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.