The toughness of a graph G is defined as the largest real number: t such that deletion of any s points from 6 results in a graph which is either connected or elst: has at most s/t components. Clearly, every hamiltonian graph is l-tough. Conversely, we conjecture that for some to, ewry to-tough graph is hamiltonian. Since a square of a k-connectc;d graph is always ktough., a proof of this conjecture with to = 2 would imply Fleischner's thec,rem (the square of a block is hamiltonian). We

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... zn infinite family of (3/2)-tough nonhamiltonian graphs. * Original version received 20 Decembeq l971; i&vised version received 29 June 1972.