Hostname: page-component-7dd5485656-hw7sx Total loading time: 0 Render date: 2025-10-31T01:41:04.571Z Has data issue: false hasContentIssue false
  • English
  • Français

RAMSEY NUMBERS OF CYCLES VERSUS MULTIPLE ODD WHEELS

Part of: Graph theory Extremal combinatorics

Published online by Cambridge University Press:  01 September 2025

ZHAOFA WANG  and
YANBO ZHANG Open the ORCID record for YANBO ZHANG [Opens in a new window]
ZHAOFA WANG
Affiliation:
School of Mathematical Sciences, Hebei Normal University , Shijiazhuang 050024, PR China and Hebei Research Center of the Basic Discipline Pure Mathematics, Shijiazhuang 050024, PR China e-mail: zfwang.edu@outlook.com
YANBO ZHANG*
Affiliation:
School of Mathematical Sciences, Hebei Normal University , Shijiazhuang 050024, PR China and Hebei Research Center of the Basic Discipline Pure Mathematics, Shijiazhuang 050024, PR China
*
e-mail: ybzhang@hebtu.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

Given two graphs G and H, the Ramsey number $R(G,H)$ is the smallest positive integer N such that every graph of order N contains G or its complement contains H as a subgraph. Let $C_n$ denote the cycle on n vertices and let $tW_{2m+1}$ denote the disjoint union of t copies of the $(2m+2)$-vertex wheel $W_{2m+1}$. We show that for integers $m\ge 1$, $t\ge 2$ and $n\ge (6m+3)t-6m+999$,

$$ \begin{align*} R(C_n, tW_{2m+1})=3n+t-3. \end{align*} $$

This result extends several previous results and settles a conjecture posed by Sudarsana [‘A note on the Ramsey number for cycle with respect to multiple copies of wheels’, Electron. J. Graph Theory Appl. 9(2) (2021), 561–566].

Keywords

Ramsey numbercyclewheel

MSC classification

Primary: 05D10: Ramsey theory
Secondary: 05C55: Generalized Ramsey theory

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 9
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

The second author was partially supported by the National Natural Science Foundation of China (NSFC) under Grant No. 11601527 and by the Natural Science Foundation of Hebei Province under Grant No. A2023205045.

References

Alweiss, R., ‘Ramsey numbers of odd cycles versus larger even wheels’, Discrete Math. 341(4) (2018), 981–989.10.1016/j.disc.2018.01.004CrossRefGoogle Scholar
Bondy, J. A. and Chvátal, V., ‘A method in graph theory’, Discrete Math. 15(2) (1976), 111–135.10.1016/0012-365X(76)90078-9CrossRefGoogle Scholar
Bondy, J. A. and Murty, U. S. R., Graph Theory, Graduate Texts in Mathematics, 244 (Springer, London, 2008).10.1007/978-1-84628-970-5CrossRefGoogle Scholar
Brandt, S., Faudree, R. J. and Goddard, W., ‘Weakly pancyclic graphs’, J. Graph Theory 27(3) (1998), 141–176.10.1002/(SICI)1097-0118(199803)27:3<141::AID-JGT3>3.0.CO;2-O3.0.CO;2-O>CrossRefGoogle Scholar
Burr, S. A. and Erdős, P., ‘Generalizations of a Ramsey-theoretic result of Chvátal’, J. Graph Theory 7(1) (1983), 39–51.10.1002/jgt.3190070106CrossRefGoogle Scholar
Burr, S. A., Erdős, P., Faudree, R. J., Rousseau, C. C. and Schelp, R. H., ‘Some complete bipartite graph-tree Ramsey numbers’, in: Graph Theory in Memory of G. A. Dirac, Annals of Discrete Mathematics, 41 (eds. Andersen, L. D., Jakobsen, I. T., Thomassen, C., Toft, B. and Vestegaard, P. D.) (North-Holland, Amsterdam, 1989), 79–89.Google Scholar
Chen, Y., Cheng, T. C. E., Miao, Z. and Ng, C. T., ‘The Ramsey numbers for cycles versus wheels of odd order’, Appl. Math. Lett. 22(12) (2009), 1875–1876.10.1016/j.aml.2009.07.014CrossRefGoogle Scholar
Chen, Y., Cheng, T. C. E., Ng, C. T. and Zhang, Y., ‘A theorem on cycle-wheel Ramsey number’, Discrete Math. 312(5) (2012), 1059–1061.10.1016/j.disc.2011.11.022CrossRefGoogle Scholar
Dirac, G. A., ‘Some theorems on abstract graphs’, Proc. Lond. Math. Soc. (3) 2 (1952), 69–81.10.1112/plms/s3-2.1.69CrossRefGoogle Scholar
Faudree, R. J. and Schelp, R. H., ‘All Ramsey numbers for cycles in graphs’, Discrete Math. 8 (1974), 313–329.10.1016/0012-365X(74)90151-4CrossRefGoogle Scholar
Greenwood, R. E. and Gleason, A. M., ‘Combinatorial relations and chromatic graphs’, Canad. J. Math. 7 (1955), 1–7.10.4153/CJM-1955-001-4CrossRefGoogle Scholar
Rosta, V., ‘On a Ramsey type problem of J. A. Bondy and P. Erdős, I, II’, J. Combin. Theory Ser. B 15 (1973), 94–120.10.1016/0095-8956(73)90035-XCrossRefGoogle Scholar
Sanhueza-Matamala, N., ‘Stability and Ramsey numbers for cycles and wheels’, Discrete Math. 339(5) (2016), 1557–1565.10.1016/j.disc.2015.12.031CrossRefGoogle Scholar
Shi, L., ‘Ramsey numbers of long cycles versus books or wheels’, European J. Combin. 31(3) (2010), 828–838.10.1016/j.ejc.2009.07.004CrossRefGoogle Scholar
Sudarsana, I. W., ‘A note on the Ramsey number for cycle with respect to multiple copies of wheels’, Electron. J. Graph Theory Appl. (EJGTA) 9(2) (2021), 561–566.10.5614/ejgta.2021.9.2.24CrossRefGoogle Scholar
Baskoro, Surahmat, E. T. and Tomescu, I., ‘The Ramsey numbers of large cycles versus wheels’, Discrete Math. 306(24) (2006), 3334–3337.Google Scholar
Baskoro, Surahmat, E. T. and Tomescu, I., ‘The Ramsey numbers of large cycles versus odd wheels’, Graphs Combin. 24(1) (2008), 53–58.Google Scholar
Zhang, L., Chen, Y. and Cheng, T. C. E., ‘The Ramsey numbers for cycles versus wheels of even order,’ European J. Combin. 31(1) (2010), 254–259.10.1016/j.ejc.2008.12.022CrossRefGoogle Scholar
Zhang, Y., Broersma, H. and Chen, Y., ‘A remark on star- ${C}_4$ and wheel- ${C}_4$ Ramsey numbers’, Electron. J. Graph Theory Appl. (EJGTA) 2(2) (2014), 110–114.10.5614/ejgta.2014.2.2.3CrossRefGoogle Scholar
Zhang, Y., Broersma, H. and Chen, Y., ‘Three results on cycle-wheel Ramsey numbers’, Graphs Combin. 31(6) (2015), 2467–2479.10.1007/s00373-014-1523-0CrossRefGoogle Scholar
Zhang, Y., Zhang, Y. and Chen, Y., ‘The Ramsey numbers of wheels versus odd cycles’, Discrete Math. 323 (2014), 76–80.10.1016/j.disc.2014.01.017CrossRefGoogle Scholar