Tough graphs and hamiltonian circuits Original Research Article

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 image components. Clearly, every hamiltonian graph is 1-tough. Conversely, we conjecture that for some image, every image-tough graph is hamiltonian. Since a square of a k-connected graph is always k-tough, a proof of this conjecture with image would imply Fleischnerʹs theorem (the square of a block is hamiltonian). We construct an infinite family of (3/2)-tough nonhamiltonian graphs.