On the size of spheres of relations with a transitive group of automorphisms

Details On the size of spheres of relations with a transitive group of automorphisms On the size of spheres of relations with a transitive group of automorphisms

2006-03-21

arxiv.org/abs/math/0603504v1

Mathematics

Combinatorics

Scientific paper

Let $\Gamma =(V,E)$ be a point-transitive reflexive relation. Let $v\in V$ and put $r=|\Gamma (v)|.$ Also assume $\Gamma ^j(v)\cap \Gamma ^{-}(v)=\{v\}$. Then $$ |\Gamma ^{j} (v)\setminus \Gamma ^{j-1} (v)| \ge r-1.$$ In particular we have $ |\Gamma ^{j} (v)| \ge 1+(r-1)j.$ The last result confirms a recent conjecture of Seymour in the case vertex-transitive graphs. Also it gives a short proof for the validity of the Caccetta-H\"aggkvist conjecture for vertex-transitive graphs and generalizes an additive result of Shepherdson.

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