Skip to main content
SpringerLink
Log in

Star-Critical Ramsey Numbers Involving Various Graphs

  • Chapter
  • First Online:
Star-Critical Ramsey Numbers for Graphs

Part of the book series: SpringerBriefs in Mathematics ((BRIEFSMATH))

  • 195 Accesses

Abstract

This chapter focuses on the evaluation of star-critical Ramsey numbers for a variety of graphs: trees, cycles, wheels, the disjoint union of complete graphs, fans, books, and their generalizations.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. J. Bondy, P. Erdős, Ramsey number for cycles in graphs. J. Combin. Theory Ser. B 14, 46–54 (1973)

    Google Scholar 

  2. M. Budden, E. DeJonge, Destroying the Ramsey property by the removal of edges. Australas. J. Combin. 74, 476–485 (2019)

    MathSciNet  MATH  Google Scholar 

  3. M. Budden, E. DeJonge, Multicolor star-critical Ramsey numbers and Ramsey-good graphs. Electron. J. Graph Theory Appl. 10, 51–66 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  4. S. Burr, Generalized Ramsey theory for graphs – a survey. In Graphs and Combinatorics (Proc. Capital Conf., George Washington Univ., Washington, D.C., 1973), Lecture Notes in Math. 406, 52–75 (1974)

    Google Scholar 

  5. S. Burr, On Ramsey numbers for large disjoint unions of graphs. Discrete Math. 70, 277–293 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. S. Burr, P. Erdős, Generalizations of a Ramsey-theoretic result of Chvátal. J. Graph Theory 7, 39–51 (1983)

    Google Scholar 

  7. S. Burr, P. Erdős, R. Faudree, C. Rousseau, R. Schelp, Some complete bipartite graph-tree Ramsey numbers. Ann. Discrete Math. 41, 79–89 (1989)

    Google Scholar 

  8. S. Burr, P. Erdős, J. Spencer, Ramsey theorems for multiple copies of graphs. Trans. Amer. Math. Soc. 209, 87–99 (1975)

    Google Scholar 

  9. S. Burr, J. Roberts, On Ramsey numbers for stars. Utilitas Math. 4, 217–220 (1973)

    MathSciNet  MATH  Google Scholar 

  10. G. Chartrand, S. Schuster, On a variation of the Ramsey number. Trans. Amer. Math. Soc. 173, 353–362 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  11. Y. Chen, T. Cheng, C. Ng, Y. Zhang, A theorem on cycle-wheel Ramsey number. Discrete Math. 312, 1059–1061 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. V. Chvátal, On Hamilton’s ideals. J. Combin. Theory Ser. B 12, 163–168 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  13. V. Chvátal, Tree-complete graph Ramsey numbers. J. Graph Theory 1, 93 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  14. V. Chvátal, F. Harary, Generalized Ramsey theory for graphs III. Small off-diagonal numbers. Pacific J. Math. 41, 335–345 (1972)

    MATH  Google Scholar 

  15. E. Cockayne, P. Lorimer, On Ramsey graph numbers for stars and stripes. Canad. Math. Bull. 18, 31–34 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  16. E. Cockayne, P. Lorimer, The Ramsey number for stripes. J. Austral. Math. Soc. (Series A) 19, 252–256 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  17. G. Dirac, Some theorems on abstract graphs. Proc. Lond. Math. Soc. 2, 68–81 (1952)

    MathSciNet  MATH  Google Scholar 

  18. P. Erdős, R. Faudree, Size Ramsey functions. In Sets, Graphs and Numbers (Budapest, 1991), Colloq. Math. Soc. Janos Bolyai 60, 219–238 (1992)

    Google Scholar 

  19. P. Erdős, R. Faudree, C. Rousseau, R. Schelp, The size Ramsey number. Period. Math. Hungar. 9, 145–161 (1978)

    Google Scholar 

  20. P. Erdős, R. Faudree, C. Rousseau, R. Schelp, On cycle-complete graph Ramsey numbers. J. Graph Theory 2, 53–64 (1978)

    Google Scholar 

  21. P. Erdős, R. Faudree, C. Rousseau, R. Schelp, The book-tree Ramsey numbers. Scientia, Ser. A, Math. Sci. 1, 111–117 (1988)

    Google Scholar 

  22. R. Faudree, S. Lawrence, T. Parsons, R. Schelp, Path-cycle Ramsey numbers. Discrete Math. 10, 269–277 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  23. R. Faudree, R. Schelp, All Ramsey numbers for cycles in graphs. Discrete Math. 8, 313–329 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  24. L. Gerencsér, A. Gyárfás, On Ramsey-type problems. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 10, 167–170 (1967)

    MathSciNet  MATH  Google Scholar 

  25. R. Greenwood, A. Gleason, Combinatorial relations and chromatic graphs. Canad. J. Math. 7, 1–7 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  26. Y. Guo, L. Volkmann, Tree-Ramsey numbers. Australas. J. Combin. 11, 169–175 (1995)

    MathSciNet  MATH  Google Scholar 

  27. S. Haghi, H. Maimani, A. Seify, Star-critical Ramsey numbers of Fn versus K4. Discrete Appl. Math. 217, 203–209 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Hamm, P. Hazelton, S. Thompson, On Ramsey and star-critical Ramsey numbers for generalized fans versus nKm. Discrete Appl. Math. 305, 64–70 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  29. Y. Hao, Q. Lin, Ramsey number of K3 versus F3,n. Discrete Appl. Math. 251, 345–348 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  30. Y. Hao, Q. Lin, Star-critical Ramsey numbers for large generalized fans and books. Discrete Math. 341, 3385–3393 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  31. F. Harary, Recent results on generalized Ramsey theory for graphs. In Graph Theory and Applications (Proc. Conf., Western Michigan Univ., 1972) Lecture Notes in Math. 303, 125–138 (1972)

    Google Scholar 

  32. F. Harary, G. Prins, Generalized Ramsey theory for graphs, IV: the Ramsey multiplicity of a graph. Networks 4, 163–173 (1974)

    MATH  Google Scholar 

  33. J. Hook, The Classification of Critical Graphs and Star-Critical Ramsey Numbers (Ph.D. Thesis, Lehigh Univ., 2010)

    Google Scholar 

  34. J. Hook, Critical graphs for R(Pn, Pm) and the star-critical Ramsey number for paths. Discuss. Math. Graph Theory 35, 689–701 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. J. Hook, G. Isaak, Star-critical Ramsey numbers. Discrete Appl. Math. 159, 328–334 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. R. Irving, Generalised Ramsey numbers for small graphs. Discrete Math. 9, 251–264 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  37. M. Jacobson, On the Ramsey multiplicity for stars. Discrete Math. 42, 63–66 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  38. M. Jacobson, On the Ramsey number for stars and a complete graph. Ars Combin. 17, 167–172 (1984)

    MathSciNet  MATH  Google Scholar 

  39. C. Jayawardene, D. Narváez, S. Radziszowski, Star-critical Ramsey numbers for cycles versus K4. Discuss. Math. Graph Theory 41, 381–390 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  40. A. Kamranian, G. Raeisi, On the star-critical Ramsey number of a forest versus complete graphs. Iran. J. Sci. Technol. Trans. Sci. 46, 499–505 (2022)

    Article  MathSciNet  Google Scholar 

  41. M. Katz, J. Reimann, An Introduction to Ramsey Theory – Fast Functions, Infinity, and Metamathematics (Student Mathematical Library, Volume 87, American Mathematical Society, Providence, 2018)

    MATH  Google Scholar 

  42. S. Lawrence, Cycle-star Ramsey numbers. Notices Amer. Math. Soc. 20, Abstract A-420 (1973)

    Google Scholar 

  43. Y. Li, C. Rousseau, Fan-complete Ramsey numbers. J. Graph Theory 23(4), 413–420 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  44. Y. Li, Y. Li, Star-critical Ramsey numbers involving large books. Discrete Appl. Math. 327, 68–76 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  45. Y. Li, Y. Li, Y. Wang, Minimal Ramsey graphs on deleting stars for generalized fans and books. Appl. Math. Comput. 372, #125006 (2020)

    Google Scholar 

  46. Y. Li, Y. Li, Y. Wang, Star-critical Ramsey numbers of generalized fans. Graphs Combin. 37, 2113–2120 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  47. Y. Li, Y. Li, Y. Wang, Star-critical Ramsey number of large cycle and book of different orders. Theoret. Comput. Sci. 866, 37–42 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  48. Y. Li, Y. Li, Y. Wang, Star-critical Ramsey numbers involving large generalized fans. Graphs Combin. 38. #130 (2022)

    Google Scholar 

  49. Z. Li, Y. Li, Some star-critical Ramsey numbers. Discrete Appl. Math. 181, 301–305 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  50. Y. Liu, Y. Chen, Star-critical Ramsey numbers of cycles versus wheels. Graphs Combin. 37, 2167–2172 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  51. Y. Liu, Y. Chen, Star-critical Ramsey numbers of wheels versus odd cycles. Acta Math. Appl. Sin. Engl. Ser. 38, 916–924 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  52. P. Lorimer, The Ramsey numbers for stripes and one complete graph. J. Graph Theory 8, 177–184 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  53. P. Lorimer, P. Mullins, Ramsey numbers for quadrangles and triangles. J. Combin. Theory Ser. B 23, 262–265 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  54. G. Omidi, G. Raeisi, A note on the Ramsey number of stars-complete graphs. Eur. J. Combin. 32, 598–599 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  55. Ø. Ore, Hamiltonian connected graphs. J. Math. Pure Appl. 42, 21–27 (1963)

    MATH  Google Scholar 

  56. T. Parsons, Path-star Ramsey numbers. J. Combin. Theory Ser. B 17, 51–58 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  57. S. Radziszowski, Small Ramsey numbers – revision 16. Electron. J. Combin. DS1.16, 116 pages (2021)

    Google Scholar 

  58. S. Radziszowski, X. Jin, Paths, cycles and wheels in graphs without antitriangles. Australas. J. Combin. 9, 221–232 (1994)

    MathSciNet  MATH  Google Scholar 

  59. C. Rousseau, J. Sheehan, A class of Ramsey problems involving trees. J. Lond. Math. Soc. 18, 392–396 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  60. A. Salman, H. Broersma, Path-fan Ramsey numbers. Discrete Appl. Math. 154, 1429–1436 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  61. S. Stahl, On the Ramsey number R(F, Km) where F is a forest. Can. J. Math. 27, 585–589 (1975)

    Article  MATH  Google Scholar 

  62. Surahmat, E. Baskoro, H. Broersma, The Ramsey numbers of fans versus K4. Bull. Inst. Combin. Appl. 43, 96–102 (2005)

    MathSciNet  MATH  Google Scholar 

  63. Y. Wang, Y. Li, Y. Li, Maximum star deleted from Ramsey graphs of book and tree. Theoret. Comput. Sci. 836, 37–41 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  64. Y. Wang, Y. Li, Y. Li, Star-critical Ramsey numbers involving graphs with long suspended paths. Discrete Math. 344. #112233 (2021)

    Google Scholar 

  65. Y. Wu, Y. Sun, S. Radziszowski, Wheel and star-critical Ramsey numbers for quadrilateral. Discrete Appl. Math. 186, 260–271 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  66. C. Xu, H. Yang, S. Zhang, On characterizing the critical graphs for matching Ramsey numbers. Discrete Appl. Math. 287, 15–20 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  67. J. Yang, Y. Huang, K. Zhang, The value of the Ramsey number R(Cn, K4) is 3(n − 1) + 1 (n ≥ 4). Australas. J. Combin. 20, 205–206 (1999)

    MathSciNet  MATH  Google Scholar 

  68. Y. Zhang, H. Broersma, Y. Chen, On star-critical and upper size Ramsey numbers. Discrete Appl. Math. 202, 174–180 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Western Carolina University, Cullowhee, NC, USA

    Mark Budden

Authors
  1. Mark Budden
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Budden, M. (2023). Star-Critical Ramsey Numbers Involving Various Graphs. In: Star-Critical Ramsey Numbers for Graphs. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-031-29981-0_2

Download citation

Publish with us

Policies and ethics

Search

Navigation