Advertisement

Combinatorica

, Volume 34, Issue 5, pp 547–559 | Cite as

The circumference of the square of a connected graph

  • Stephan Brandt
  • Janina Müttel
  • Dieter Rautenbach
Original Paper
  • 145 Downloads

Abstract

The celebrated result of Fleischner states that the square of every 2-connected graph is Hamiltonian. We investigate what happens if the graph is just connected. For every n ≥ 3, we determine the smallest length c(n) of a longest cycle in the square of a connected graph of order n and show that c(n) is a logarithmic function in n. Furthermore, for every c ≥ 3, we characterize the connected graphs of largest order whose square contains no cycle of length at least c.

Mathematics Subject Classification (2000)

05C38 05C76 05C05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R. Diestel: Graph Theory, Springer, 2010.CrossRefGoogle Scholar
  2. [2]
    H. Fleischner: The square of every two-connected graph is Hamiltonian, J. Comb. Theory, Ser. B 16 (1974), 29–34.CrossRefMATHMathSciNetGoogle Scholar
  3. [3]
    H. Fleischner: In the square of graphs, Hamiltonicity and pancyclicity, Hamiltonian connectedness and panconnectedness are equivalent concepts, Monatsh. Math. 82 (1976), 125–149.CrossRefMATHMathSciNetGoogle Scholar
  4. [4]
    M. Ch. Golumbic: Algorithmic graph theory and perfect graphs, Elsevier, Amsterdam, 2004.MATHGoogle Scholar
  5. [5]
    F. Harary and A. Schwenk: Trees with Hamiltonian square, Mathematika, Lond. 18 (1971), 138–140.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Stephan Brandt
    • 1
  • Janina Müttel
    • 2
  • Dieter Rautenbach
    • 2
  1. 1.Department of Mathematics and Computer Science (IMADA)University of Southern DenmarkOdenseDenmark
  2. 2.Institut für Optimierung und Operations ResearchUniversität UlmUlmGermany

Personalised recommendations