On Star-critical (K1,n,K1,m + e) Ramsey numbers

Let \(G, H\) be finite graphs without loops or multiple edges and \(K_n\) denote the complete graph on \(n\) vertices. If for every red/blue colouring of edges of the complete graph \(K_n\), there exists a red copy of \(G\), or a blue copy of \(H\), we will say that \(K_n\rightarrow (G,H)\). The Ram...

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Bibliographic Details
Published in:arXiv.org 2020-09
Main Authors: Jayawardene, C J, Senadheera, J N, K A S N Fernando, Navaratna, W C W
Format: Article
Language:English
Subjects:
Apexes
Coloring
Graph theory
Numbers
Online Access:Get full text
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Summary:Let \(G, H\) be finite graphs without loops or multiple edges and \(K_n\) denote the complete graph on \(n\) vertices. If for every red/blue colouring of edges of the complete graph \(K_n\), there exists a red copy of \(G\), or a blue copy of \(H\), we will say that \(K_n\rightarrow (G,H)\). The Ramsey number \(r(G, H)\) is defined as the smallest positive integer \(n\) such that \(K_{n} \rightarrow (G, H)\). Star-critical Ramsey number \(r_*(G, H)\) is defined as the largest value of \(k\) such that \(K_{r(G,H)-1} \sqcup K_{1,k} \rightarrow (G, H)\). In this paper, we will find \(r_*(K_{1,n}, K_{1,m}+e)\) for all \(n,m \geq 3\).
ISSN:2331-8422