A sufficient condition for connected Nz-locally connected K-1,K-3-free graphs without vertices of degree 1 to be Hamiltonian is given in terms of subgraphs and their local properties. This condition is weaker than that given by Ryjacek (J. Graph Theory 14 (3) (1990) 321-331). (C) 2000 Elsevier Science B.V. All rights reserved. MSG: 05C38: 05C45.