Abstract
The toughness of a graph G is defined as the largest real number t such that deletion of any s points from G results in a graph which is either connected or else has at most s/ t components. Clearly, every hamiltonian graph is 1-tough. Conversely, we conjecture that for some t 0, every t 0-tough graph is hamiltonian. Since a square of a k-connected graph is always k-tough, a proof of this conjecture with t 0 = 2 would imply Fleischner's theorem (the square of a block is hamiltonian). We construct an infinite family of ( 3 2 )-tough nonhamiltonian graphs.
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