Skip to main content

Cycles in abelian cayley graphs with a proscribed vertex

  • 474 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 1073)

Abstract

A graph Γ has the property C(m,n) if, whenever M, N are disjoint sets of vertices of Γ with |M|=m and |N|=n, there exists a cycle in Γ which includes all the vertices of M and which avoids all the vertices of N. Let G be a group generated by a subset X of G. We denote by G(X) the graph whose vertices are elements of G and two vertices a, b are adjacent if and only if a−1 e X U X−1. The graph G(X) is known as a Cayley graph. For an abelian group G of order p, G(X) has the property C(p−1,1) if and only if G(X) is neither cyclic nor bipartite, which in turn, is equivalent to G(X) being hamilton-connected. Moreover, if G(X) is bipartite of order 2t but not cyclic, then G(X) has the property C(t−1,1) but not C(t,1). If G(X) is cyclic then it is not C(m,1) for any m.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   46.00
Price excludes VAT (Canada)
  • 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D. A. Holton and M. D. Plummer, Cycles through prescribed and forbidden sets, Ann. of Discrete Math. 16 (1982), 129–147.

    MathSciNet  MATH  Google Scholar 

  2. A. Gardiner and D. A. Holton, Cycles with prescribed and proscribed vertices, University of Melbourne, Mathematics Research Report No.6, 1981.

    Google Scholar 

  3. D. A. Holton, B. D. McKay, M. D. Plummer and C. Thomassen, A nine point theorem for 3-connected cubic graphs, Combinatorica 2(1) (1982), 53–62.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. C. C. Chen and N. Quimpo, On strongly hamiltonian abelian group graphs, Combinatorial mathematics VIII, Lecture Notes Math. 884, (1981), 23–34.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. C. C. Chen and N. Quimpo, Hamiltonian Cayley graphs of order pq (to appear).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1984 Springer-Verlag

About this paper

Cite this paper

Chen, C.C., Holton, D.A. (1984). Cycles in abelian cayley graphs with a proscribed vertex. In: Koh, K.M., Yap, H.P. (eds) Graph Theory Singapore 1983. Lecture Notes in Mathematics, vol 1073. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0073102

Download citation

  • DOI: https://doi.org/10.1007/BFb0073102

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-13368-1

  • Online ISBN: 978-3-540-38924-8

  • eBook Packages: Springer Book Archive