The circumference of the square of a connected graph

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.

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References

  1. [1]

    R. Diestel: Graph Theory, Springer, 2010.

    Book  Google Scholar 

  2. [2]

    H. Fleischner: The square of every two-connected graph is Hamiltonian, J. Comb. Theory, Ser. B 16 (1974), 29–34.

    Article  MATH  MathSciNet  Google 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.

    Article  MATH  MathSciNet  Google Scholar 

  4. [4]

    M. Ch. Golumbic: Algorithmic graph theory and perfect graphs, Elsevier, Amsterdam, 2004.

    MATH  Google Scholar 

  5. [5]

    F. Harary and A. Schwenk: Trees with Hamiltonian square, Mathematika, Lond. 18 (1971), 138–140.

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Stephan Brandt.

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Brandt, S., Müttel, J. & Rautenbach, D. The circumference of the square of a connected graph. Combinatorica 34, 547–559 (2014). https://doi.org/10.1007/s00493-014-2899-4

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Mathematics Subject Classification (2000)

  • 05C38
  • 05C76
  • 05C05