On zero-sum Ramsey numbers--stars
type: Item
instance of: scholarly article
title: On zero-sum Ramsey numbers--stars
zbMATH Open document ID: 0771.05068
DOI: 10.1016/0012-365X(92)90621-L
author: Yair Caro
published in: Discrete Mathematics
publication date: 1993-01-16T00:00:00Z
review text: For \(2\leq k\leq n\), \(R(K_{1,n},\mathbb{Z}_ k)\) is the minimal integer \(t\) such that for every \(\mathbb{Z}_ k\)-coloring of the edges of \(K_ t\), there is a copy of \(K_{1,n}\) such that the sum of the colors of the edges is \(0\in\mathbb{Z}_ k\). The author generalizes a result of \textit{A. Bialostocki} and \textit{P. Dierker} [Discrete Math. 110, No. 1-3, 1-8 (1992)] by proving \[ R(K_{1,n},\mathbb{Z}_ k)=\begin{cases} n=k-1 & n\equiv k\equiv 0\bmod k \\ n+k & \text{otherwise.}\end{cases} \] Also, a directed version of the generalized Ramsey number \(R\), denoted by \(R^*\), is considered. For \(n\geq k\geq 2\) and \(k| n\), it is shown that \(R^*(K_{1,n},\mathbb{Z}_ k)=2(n+k-1)\), where \(R(K_{1,n},\mathbb{Z}_ k)\) is the minimal integer \(t\) such that for every \(\mathbb{Z}_ k\)-coloring of the edges of \(K_ t\) and every orientation of the edges, there is a directed copy of \(K_{1,n}\) such that the sum of the colors of the edges is \(0\in\mathbb{Z}_ k\).
reviewed by: Ralph Faudree
Mathematics Subject Classification ID: 05C55
Mathematics Subject Classification ID: 05C38
Mathematics Subject Classification ID: 05C70
zbMATH DE Number: 89877
zbMATH Keywords: zero-sum
zbMATH Keywords: stars
zbMATH Keywords: coloring
zbMATH Keywords: Ramsey number
MaRDI profile type: Publication
cites work: On the Erdős-Ginzburg-Ziv theorem and the Ramsey numbers for stars and matchings
cites work: Q3470501
cites work: Q3490024
cites work: On zero sum Ramsey numbers: Multiple copies of a graph
cites work: Q4044600
cites work: Q5841589
cites work: Parallel concepts in graph theory
description: scientific article; zbMATH DE number 89877