^{1} Department of Mathematics and Computer Science (IMADA), Faculty of Science, SDU^{2} unknown^{3} Department of Mathematics and Computer Science (IMADA), Faculty of Science, SDU

DOI:

10.1007/s00493-014-2899-4

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 a parts per thousand yen 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 a parts per thousand yen 3, we characterize the connected graphs of largest order whose square contains no cycle of length at least c.