Abstract
Let \(\mathcal {R}(G,H)\) denote the set of all graphs F satisfying \(F \rightarrow (G,H)\) and for every \(e \in E(F),\) \((F-e) \nrightarrow (G,H).\) In this paper, we derive the necessary and sufficient conditions for graphs belonging to \(\mathcal {R}(mK_2,H)\) for any graph H and each positive integer m. We give all disconnected graphs in \(\mathcal {R}(mK_2,H),\) for any connected graph H. Furthermore, we prove that if \(F \in \mathcal {R}(mK_2,P_3),\) then any graph obtained by subdividing one non-pendant edge in F will be in \(\mathcal {R}((m+1)K_2,P_3)\).
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Acknowledgements
This research was supported by Research Grant “Program Hibah Riset Unggulan ITB-DIKTI”, Ministry of Research, Technology and Higher Education, Indonesia. The first author would like to acknowledge Prof. Zdeněk Ryjáček for providing support during her research visit of two months durations in October - November 2014 at University of West Bohemia in Pilsen.
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Wijaya, K., Baskoro, E.T., Assiyatun, H. et al. On Ramsey \((mK_2,H)\)-Minimal Graphs. Graphs and Combinatorics 33, 233–243 (2017). https://doi.org/10.1007/s00373-016-1748-1
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DOI: https://doi.org/10.1007/s00373-016-1748-1