Abstract
For two graphs G and H, the Ramsey number R(G, H) is the smallest integer n such that for any n-vertex graph, either it contains G or its complement contains H. Let \(S_{n}\) be a star of order n and \(W_{s,m}\) be a generalised wheel \(K_{s}\vee C_{m}.\) Previous studies by Wang and Chen (Graphs Comb 35(1):189–193, 2019) and Chng et al. (Discret Math 344(8):112440, 2021) imply that a tree is \(W_{s,4}\)-good, \(W_{s,5}\)-good, \(W_{s,6}\)-good, and \(W_{s,7}\)-good for \(s\geqslant 2.\) In this paper, we study the Ramsey numbers \(R(S_{n}, W_{s,8}),\) and our results indicate that trees are not always \(W_{s,8}\)-good.
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References
Baskoro, E.T., Surahmat, S., Nababan, S.M., Miller, M.: On Ramsey numbers for trees versus wheels of five or six vertices. Graphs Combin. 18(4), 717–721 (2002)
Bondy, J.A.: Pancyclic graphs I. J. Combin. Theory Ser. B 11(1), 80–84 (1971)
Brandt, S., Faudree, R., Goddard, W.: Weakly pancyclic graphs. J. Graph Theory 27(3), 141–176 (1998)
Burr, S.A.: Ramsey numbers involving graphs with long suspended paths. J. Lond. Math. Soc. 24(3), 405–413 (1981)
Chen, Y., Zhang, Y., Zhang, K.: The Ramsey numbers of stars versus wheels. Eur. J. Combin. 25(7), 1067–1075 (2004)
Chng, Z., Tan, T., Wong, K.: On the Ramsey numbers for the tree graphs versus certain generalised wheel graphs. Discret. Math. 344(8), 112440 (2021)
Dirac, G.A.: Some theorems on abstract graphs. Proc. Lond. Math. Soc. 2, 69–81 (1952)
Hajnal, A., Szemerédi, E.: Proof of a conjecture of P. Erdős. In: Combinatorial Theory and Its Applications, II (Proc. Colloq., Balatonfured, 1969), pp. 601–623 (1970)
Hasmawati, E.: Bilangan Ramsey: untuk kombinasi Graf Bintang terhadap Graf Roda. Tesis Magister, Departemen Matematika ITB, Indonesia (2004)
Hasmawati, E., Baskoro, T., Assiyatun, H.: Star-wheel Ramsey numbers. J. Comb. Math. Combin. Comput. 55, 123–128 (2005)
Li, B., Schiermeyer, I.: On star-wheel Ramsey numbers. Graphs Combin. 32(2), 733–739 (2016)
Lin, Q., Li, Y., Dong, L.: Ramsey goodness and generalized stars. Eur. J. Combin. 31(5), 1228–1234 (2010)
Surahmat, S., Baskoro, E.T.: On the Ramsey number of a path or a star versus \(W_4\) or \(W_5\). In: Proceedings of the 12-th Australasian Workshop on Combinatorial Algorithms (Bandung, Indonesia), July 14–17, pp. 174–178 (2001)
Wang, L., Chen, Y.: The Ramsey numbers of trees versus generalized wheels. Graphs Combin. 35(1), 189–193 (2019)
Zhang, Y., Chen, Y., Zhang, K.: The Ramsey numbers for stars of even order versus a wheel of order nine. Eur. J. Combin. 29(7), 1744–1754 (2005)
Zhang, Y., Cheng, T.C.E., Chen, Y.: The Ramsey numbers for stars of odd order versus a wheel of order nine. Discret. Math. Algorithms Appl. 1(3), 413–436 (2009)
Acknowledgements
We are grateful to reviewers for checking all the details and giving us valuable comments to help improve the presentation. The research is supported in part by the National Natural Science Foundation of China (no. 11931002).
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This work was supported by the NSFC (Grant no. 11931002).
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Zhang, Y., Peng, Y. Ramsey Numbers of Stars Versus Generalised Wheels. Commun. Appl. Math. Comput. (2024). https://doi.org/10.1007/s42967-023-00316-3
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DOI: https://doi.org/10.1007/s42967-023-00316-3