Skip to main content
SpringerLink
Log in

Structural Properties and Hamiltonicity of Neighborhood Graphs

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

Let G = (V, E) be a simple undirected graph. N(G) =  (V, E N ) is the neighborhood graph of the graph G, if and only if

$$E_N = \{\{a,b\}\,|\, a \neq b\,\wedge\,\exists\, x \, \in V: \{x,a\} \in E \, \wedge \, \{x,b\} \in E \}.$$

We present several structural properties of N(G) and characterize the hamiltonicity of N(G) by means of chords of a hamiltonian cycle in G.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Acharya, B.D., Vartak, M.N.: Open neighborhood graphs. Indian Institute of Technology, Department of Mathematics. Research Report No. 7, Bombay (1973)

  2. Boland J.W., Brigham R.C., Dutton R.D.: Embedding arbitrary graphs in neighborhood graphs. J. Comb. Inf. Syst. Sci. 12, 101–112 (1987)

    MATH  MathSciNet  Google Scholar 

  3. Boland J.W., Brigham R.C., Dutton R.D.: The difference between a neighborhood graph and a wheel. Congr. Numerantium 58, 151–156 (1987)

    MathSciNet  Google Scholar 

  4. Bondy J.A.: Pancyclic graphs. J. Comb. Theor. Ser. B 11, 80–84 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  5. Brigham R.C., Dutton R.D.: On neighborhood graphs. J. Comb. Inf. Syst. Sci. 12, 75–85 (1987)

    MATH  MathSciNet  Google Scholar 

  6. Diestel R.: Graph Theory, 2nd edn. Springer, Berlin (2000)

    Google Scholar 

  7. Exoo G., Harary F.: Step graphs. J. Comb. Inf. Syst. Sci. 5, 52–53 (1980)

    MATH  MathSciNet  Google Scholar 

  8. Fraughnaugh K.F., Lundgren J.R., Merz S.K., Maybee J.S., Pullman N.J.: Competition graphs of strongly connected and hamiltonian digraphs. SIAM. J. Discret. Math. 8, 179–185 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  9. Greenberg H.J., Lundgren J.R., Maybee J.S.: The inversion of 2-step graphs. J. Comb. Inf. Syst. Sci. 8, 33–43 (1983)

    MATH  MathSciNet  Google Scholar 

  10. Kim, S.R.: The competition number and its variants. In: Gimbel, J., Kennedy, J.W., Quintas, L.V. (eds.) Quo Vadis, Graph Theory? Ann. Discret. Math. 55, 313–326 (1993)

  11. Lundgren J.R.: Food webs, competition graphs, competition-common enemy graphs and niche graphs. In: Roberts, F. (eds) Applications of Combinatorics and Graph Theory to the Biological and Social Sciences, IMA, vol. 17, pp. 221–243. Springer, New York (1989)

    Google Scholar 

  12. Lundgren J.R., Merz S.K., Maybee J.S., Rasmussen C.W.: A characterization of graphs with internal two-step graphs. Linear Algebra Appl. 217, 203–223 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  13. Lundgren J.R., Merz S.K., Rasmussen C.W.: Chromatic numbers of competition graphs. Linear Algebra Appl. 217, 225–239 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  14. Lundgren J.R., Rasmussen C.: Two-step graphs of trees. Discret. Math. 119, 123–139 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  15. Lundgren J.R., Rasmussen C.W., Maybee J.S.: Interval competition graphs of symmetric digraphs. Discret. Math. 119, 113–122 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  16. Miller M.M., Brigham R.C., Dutton R.D.: An equation involving the neighborhood (two step) and line graphs. Ars. Comb. 52, 33–50 (1999)

    MATH  MathSciNet  Google Scholar 

  17. Ore O.: A note on hamiltonian circuits. Am. Math. Mon. 67, 55 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  18. Pfützenreuter, M.: Konkurrenzgraphen von ungerichteten Graphen. Bachelor thesis, University of Lübeck, Lübeck (2006)

  19. Roberts, F.S.: Competition graphs and phylogeny graphs. In: Lovász L. (ed.) Graph Theory and Combinatiorial Biology. Proceedings of International Colloquium Balatonlelle, 1996, Bolyai Society of Mathematical Studies, vol. 7, pp. 333–362. Budapest (1999)

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Freiberg University of Mining and Technology, Prüferstr. 1, 09596, Freiberg, Germany

    Ingo Schiermeyer & Martin Sonntag

  2. Institute of Mathematics, University of Lübeck, Wallstr. 40, 23560, Lübeck, Germany

    Hanns-Martin Teichert

Authors
  1. Ingo Schiermeyer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Sonntag
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hanns-Martin Teichert
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin Sonntag.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schiermeyer, I., Sonntag, M. & Teichert, HM. Structural Properties and Hamiltonicity of Neighborhood Graphs. Graphs and Combinatorics 26, 433–456 (2010). https://doi.org/10.1007/s00373-010-0909-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-010-0909-x

Keywords

Mathematics Subject Classification (2000)

Search

Navigation