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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...
Dedicated to the memory of Paul Erds In [9] Thomassen proved that a -connected graph either contains k vertex disjoint odd cycles or an odd cycle cover containing at most 2k-2 vertices, i.e. he showed that the ErdsPsa property holds for odd cycles in highly connected graphs. In this paper, we will show that the above statement is still valid for 576k-connected graphs which is essentially best possible.
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