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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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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DOI: https://doi.org/10.1007/s00493-014-2899-4