Connected Size Ramsey Numbers for The Pair Complete Graph of Order Two versus Union Complete Graph of Order Three
DOI:
https://doi.org/10.20956/j.v21i2.41850Keywords:
graph; connected; size Ramsey number; complete graph., connected, size Ramsey number, complete graphAbstract
Let F,G, and H be finite, simple, and undirected graphs. The connected size Ramsey number r ̂_c (G,H) of graph G and H is the least integer k such that there is a connected graph F with k edges and if the edge set of F is arbitrarily colored by red or blue, then there always exists either a red copy of G or a blue copy of H. This paper shows that the connected size Ramsey number r ̂_c (2K_2,〖nK〗_3 )=4n+3, for n≥4.References
[1] Assiyatun, H., Rahajeng, B. & Baskoro, E. T., 2019. The Connected Size Ramsey Numbers for Matchings Versus Small Disconnected Graphs. Electronic Journal of Graph Theory and Applications, Vol. 7, No. 1, 113 – 119.
[2] Draganaic ́, N. & Petrova, K., 2022. Size-Ramsey Numbers of Graphs with Maximum Degree Three, arXiv: 2207.05048 [math.CO]
[3] Erdo ̈s, P., Faudree, R.J., Rossseau, C.C.& Schelp, R.H., 1978. Size Ramsey Number. Period. Math. Hungar, Vol. 9, No. 1, 145 – 161.
[4] Faudree, R.J., Rousseau C.C. & Sheehan, J., 1983. A Class of size Ramsey Problems Involving Stars, In: Bollobas B, editor. Graph Theory and Combinatorics. Cambiridge: Cambiridge Univ Press, 273 – 281.
[5] Faudree, R.J. & Sheehan, J., 1983. Size Ramsey Numbers for Small-Order Graphs, J. Graph Theory, Vol. 7, No. 1, 53 – 55.
[6] Indrayani, S., Hasmawati & Nurdin., 2017. Determination of Connected Size Ramsey Numbers for Pair Complete Graph of Order Two Versus Union of Several Complete Graphs of Order Three, Prosiding Senamas 2017, 99 – 108.
[7] Javadi, R., Khoeini, F., Omidi, R.G. & Pokrovskiy, A., 2017. On The Size-Ramsey Number of Cycles, arXiv: 1701.07348v1 [math.CO].
[8] Lortz, R. & Mengersen, I., 1998. Size Ramsey Results for Paths Versus Stars. Australas J. Combin., Vol 18, No. 1, 3 – 12.
[9] Rahajeng, B., Baskoro, E. T. & Assiyatun, H., 2016. Connected Size Ramsey Numbers for Matchings and Stars. AIP Conf. Proc., 1707, 020015; doi:10.1063/1.4940816.
[10] Rahajeng, B., Baskoro, E. T. & Assiyatun, H., 2017. Connected Size Ramsey Numbers for Matchings Versus Small Stars or Cycles. Proc. Indian Acad. Sci. Math. Sci, Vol. 127, No. 5, 787 – 792.
[11] Rahajeng, B., Baskoro, E. T. & Assiyatun, H., 2015. Connected Size Ramsey Numbers for Matchings Versus Cycles or Paths. Procedia Computer Science, Vol. 74, No. 1, 32 – 37.
[12] Wang, S., Song R., Zhang, Y. & Zhang, Y., 2022. Connected Size Ramsey Numbers of Matchings Versus A Small Path or Cycle, arXiv: 2205.03965v1 [math.CO].
[13] Silaban, D. R., Baskoro, E.T. & Uttunggadewa, S., 2018). Restricter Size Ramsey Numbers for 2K_2 Versus Dense Connected Graphs of Order Six, IOP Conf. Series: Journal of Physics 1008, 012034, doi:10.1088/1742-6596/1008/1/012034.
[14] Vito, V., Nabila A. C., Safitri, E. & Silaban, D. R., 2021. The Size Ramsey and Connected Size Ramsey Numbers for Matchings Versus Path. Journal of Physic: Conference Series, 1725(2021) 012098, doi:10.1088/1742-6596/1725/1/012098
[15] Li, X., Fahad, A., Zhou, X. & Yang, H., 2020. Exact Values For Some Size Ramsey Numbers of Paths and Cycles. Frontiers in Physics, Vol 8, No. 350, 1 – 4.
[16] Zhang, Y., & Zhi, H., 2023. A Proof Of Conjecture On Matching-Path Connected Size Ramsey Numbers. AIMS Mathematics, Vol. 8, No. 4, 8027 – 8033.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2025 Jurnal Matematika, Statistika dan Komputasi

This work is licensed under a Creative Commons Attribution 4.0 International License.
This work is licensed under a Creative Commons Attribution 4.0 International License.
Jurnal Matematika, Statistika dan Komputasi is an Open Access journal, all articles are distributed under the terms of the Creative Commons Attribution License, allowing third parties to copy and redistribute the material in any medium or format, transform, and build upon the material, provided the original work is properly cited and states its license. This license allows authors and readers to use all articles, data sets, graphics and appendices in data mining applications, search engines, web sites, blogs and other platforms by providing appropriate reference.