Proceedings - Mathematical Sciences

, Volume 127, Issue 5, pp 787–792 | Cite as

Connected size Ramsey number for matchings vs. small stars or cycles

  • Budi Rahadjeng
  • Edy Tri Baskoro
  • Hilda Assiyatun
Article
  • 79 Downloads

Abstract

The notation \(F\rightarrow (G,H)\) means that if the edges of F are colored red and blue, then the red subgraph contains a copy of G or the blue subgraph contains a copy of H. The connected size Ramsey number \(\hat{r}_c(G,H)\) of graphs G and H is the minimum size of a connected graph F satisfying \(F\rightarrow (G,H)\). For \(m \ge 2,\) the graph consisting of m independent edges is called a matching and is denoted by \(mK_2\). In 1981, Erdös and Faudree determined the size Ramsey numbers for the pair \((mK_2, K_{1,t})\). They showed that the disconnected graph \(mK_{1,t} \rightarrow (mK_2,K_{1,t})\) for \( t,m \ge 1\). In this paper, we will determine the connected size Ramsey number \(\hat{r}_c(nK_2, K_{1,3})\) for \(n\ge 2\) and \(\hat{r}_c(3K_2, C_4)\). We also derive an upper bound of the connected size Ramsey number \(\hat{r}_c(nK_2, C_4),\) for \(n\ge 4\).

Keywords

Connected size Ramsey number cycle matching star 

2010 Mathematics Subject Classification

05D10 05C55 

Notes

Acknowledgements

This research was supported by Research Grant “Program Riset dan Inovasi KK-ITB”, Ministry of Research, Technology and Higher Education, Indonesia.

References

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Copyright information

© Indian Academy of Sciences 2017

Authors and Affiliations

  1. 1.Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural SciencesInstitut Teknologi Bandung (ITB)BandungIndonesia

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