Barycentric and Zero-Sum Ramsey Numbers

Abstract

Let $G$ be a finite abelian group of order $n$. The barycentric Ramsey number $BR(H,G)$ is the minimum positive integer $r$ such that any coloring of the edges of the complete graph $K_r$ by elements of $G$ contains a subgraph $H$ whose assigned edge colors constitute a barycentric sequence, i.e., there exists one edge whose color is the “average” of the colors of all edges in $H$. When the number of edges $e(H)\equiv 0(\mathrm{mod}\exp (G))$, $BR(H,G)$ are the well-known zero-sum Ramsey numbers $R(H,G)$. In this work, these Ramsey numbers are determined for some graphs, in particular, for graphs with five edges without isolated vertices using $G={\mathbb{Z}}_n$, where $2\le n\le 4$, and for some graphs $H$ with $e(H)\equiv 0(\mathrm{mod}2)$ using $G={\mathbb{Z}}_2^s$.