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Proceedings - Mathematical Sciences

, Volume 127, Issue 5, pp 779–786 | Cite as

Some infinite families of Ramsey (\({\varvec{P}}_\mathbf{3},{\varvec{P}}_{{\varvec{n}}}\))-minimal trees

  • D RahmadaniEmail author
  • E T Baskoro
  • M Bača
  • H Assiyatun
  • A Semaničová-Feňovčíková
Article

Abstract

For any given two graphs G and H, the notation \(F\rightarrow \) (GH) means that for any red–blue coloring of all the edges of F will create either a red subgraph isomorphic to G or a blue subgraph isomorphic to H. A graph F is a Ramsey (GH)-minimal graph if \(F\rightarrow \) (GH) but \(F-e\nrightarrow (G,H)\), for every \(e \in E(F)\). The class of all Ramsey (GH)-minimal graphs is denoted by \(\mathcal {R}(G,H)\). In this paper, we construct some infinite families of trees belonging to \(\mathcal {R}(P_3,P_n)\), for \(n=8\) and 9. In particular, we give an algorithm to obtain an infinite family of trees belonging to \(\mathcal {R}(P_3,P_n)\), for \(n\ge 10\).

Keywords

Ramsey minimal graph coloring Ramsey infinite tree 

2010 Mathematics Subject Classification

Primary: 05C55 Secondary: 05D10 

Notes

Acknowledgements

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

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Copyright information

© Indian Academy of Sciences 2017

Authors and Affiliations

  • D Rahmadani
    • 1
    Email author
  • E T Baskoro
    • 1
  • M Bača
    • 2
  • H Assiyatun
    • 1
  • A Semaničová-Feňovčíková
    • 2
  1. 1.Combinatorial Mathematics Research Group, Department of Mathematics, Faculty of Mathematics and Natural SciencesInstitut Teknologi Bandung (ITB)BandungIndonesia
  2. 2.Department of Applied Mathematics and InformaticsTechnical UniversityKosiceSlovak Republic

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