World Scientific
  • Search
Skip main navigation

Cookies Notification

We use cookies on this site to enhance your user experience. By continuing to browse the site, you consent to the use of our cookies. Learn More
×
Our website is made possible by displaying certain online content using javascript.
In order to view the full content, please disable your ad blocker or whitelist our website www.worldscientific.com.

System Upgrade on Tue, Oct 25th, 2022 at 2am (EDT)

Existing users will be able to log into the site and access content. However, E-commerce and registration of new users may not be available for up to 12 hours.
For online purchase, please visit us again. Contact us at [email protected] for any enquiries.

Uniformly Connected Graphs — A Survey

    This article is part of the issue:

    A graph G of order n2 is k-uniformly connected for an integer k with 1kn1 if for every pair u, v of distinct vertices of G, there is a uv path of length k. A number of results, conjectures, and problems are presented concerning k-uniformly connected graphs for various integers k. These include the special cases where k=2 and k=3. Graphs are discussed that are k-uniformly connected for a particular integer k but are not j-uniformly connected for every integer jk. Also, graphs are considered in which there is a unique uv path of length k for a particular value of k. Sets S of positive integers are considered for which there exists a graph G such that G is k-uniformly connected if and only if kS.

    PACS: AMS Subject Classification: 05C38

    References

    • 1. M. Aigner and G. Ziegler, Proofs from The Book, 5th edn. (Springer-Verlag, Berlin, 2014). Google Scholar
    • 2. N. Almohanna, D. Olejniczak and P. Zhang, Hamiltonian-connected graphs with additional properties, Congr. Numer. 231 (2018) 291–302. Google Scholar
    • 3. G. Chartrand, D. Olejniczak and P. Zhang, Uniformly connected graphs, Ars Combin. To appear. Google Scholar
    • 4. G. Chartrand and P. Zhang, A First Course in Graph Theory (Dover, New York, 2012). Google Scholar
    • 5. G. A. Dirac, Some theorems on abstract graphs, Proc. London Math. Soc. 2 (1952) 69–81. CrossrefGoogle Scholar
    • 6. P. Erdős, A. Rényi and V. T. Sós, On a problem of graph theory, Studia Sci. Math. Hungar. (1966) 215–235. Google Scholar
    • 7. R. J. Faudree and R. H. Schelp, Path connected graphs, Acta Math. Hung. 25 (1974) 313–319. CrossrefGoogle Scholar
    • 8. F. Fujie and P. Zhang, Covering Walks in Graphs (Springer, New York, 2014). CrossrefGoogle Scholar
    • 9. R. J. Gould, Updating the Hamiltonian problem — A survey, J. Graph Theory 15 (1991) 121–157. Crossref, ISIGoogle Scholar
    • 10. R. J. Gould, Advances on the Hamiltonian problem — A survey, Graphs Combin. 15 (2003) 7–52. CrossrefGoogle Scholar
    • 11. R. J. Gould, Recent advances on the Hamiltonian problem — Survey III, Graphs Combin. 30 (2014) 1–42. Crossref, ISIGoogle Scholar
    • 12. O. Ore, Hamilton connected graphs, J. Math. Pures Appl. 42 (1963) 21–27. Google Scholar
    • 13. D. Watts, Six Degrees: The Science of a Connected Age (W. W. Norton, New York, 2003). Google Scholar
    • 14. J. E. Williamson, On Hamiltonian-Connected Graphs, Ph.D. Dissertation, Western Michigan University (1973). Google Scholar
    • 15. The six degrees of separation, Exploring Your Mind (April 24, 2019). Google Scholar