Skip to main content
SpringerLink
Log in

Locating Pairs of Vertices on a Hamiltonian Cycle in Bigraphs

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

Abstract

Let G be a simple \(m\times m\) bipartite graph with minimum degree \(\delta (G)\ge m/2+1\). We prove that for every pair of vertices x, y, there is a Hamiltonian cycle in G such that the distance between x and y along that cycle equals k, where \(2\le k<m/6\) is an integer having appropriate parity. We conjecture that this is also true up to \(k\le m\).

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Ash, P., Jackson, B.: Dominating cycles in bipartite graphs. In: Bondy, J.A., Murty, U.S.R. (eds.) Progress in Graph Theory, pp. 81–87. Academic Press, New York (1984)

    Google Scholar 

  2. Hui, D., Faudree, R., Lehel, J., Yoshimoto, K.: A panconnectivity theorem for bipartite graphs (submitted)

  3. Faudree, R., Li, Hao: Locating pairs of vertices on a Hamiltonian cycle. Discrete Math. 321, 2700–2706 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Faudree, R., Lehel, J., Yoshimoto, K.: Note on locating pairs of vertices on a Hamiltonian cycle. Graphs Combin. 30, 887–894 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Jackson, B.: Long cycles in bipartite graphs. J. Combin. Theory, Ser. B 38, 118–131 (1985)

  6. Nash-Williams, C.St.J.A.: Edge-disjoint Hamiltonian circuits in graphs with vertices of large valency. In: Studies in Pure Mathematics. Academic Press, London, pp. 157–183 (1971)

  7. Whitney, H.: Congruent graphs and the connectivity of graphs. Am. J. Math. 54, 150–168 (1932)

    Article  MathSciNet  MATH  Google Scholar 

  8. Williamson, J.: Panconnected graphs II. Period. Math. Hungar. 8, 105–116 (1977)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The University of Memphis, Memphis, TN, 38152, USA

    Ralph J. Faudree

  2. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary

    Jeno Lehel

  3. Nihon University, Tokyo, 101-8308, Japan

    Kiyoshi Yoshimoto

Authors
  1. Ralph J. Faudree
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jeno Lehel
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kiyoshi Yoshimoto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jeno Lehel.

Additional information

This work was supported by JSPS KAKENHI Grant Number 26400190.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Faudree, R.J., Lehel, J. & Yoshimoto, K. Locating Pairs of Vertices on a Hamiltonian Cycle in Bigraphs. Graphs and Combinatorics 32, 963–986 (2016). https://doi.org/10.1007/s00373-015-1626-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-015-1626-2

Keywords

Mathematics Subject Classification

Search

Navigation