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… Expand