Vertex-cover in random graphs with small connectivity: an exact solution
  This paper has been withdrawn by the author, due to the fact that the main result in it has already been obtained in [1] for any c < e, see also [2] and [3].   Moreover the formula which gives the minimal vertex-cover in a tree   (see the abstract) has already been derived in [4]. I thank M. Bauer, O. Golinelli, F. Ricci-Tersenghi, G. Semerjian and M. Weigt for having brought to my attention [1] and M.B. and O.G. for [4].   [1] M. Bauer and O. Golinelli, Eur. Phys. J. B 24, 339-352 (2001);   [2] R. M. Karp and M. Sipser, Proc. 22nd IEEE Symposium on Foundations of Computing,(1981), 364-375;   [3] J. Aronson, A. Frieze, and B.G. Pittel, Random Structures and Algorithms 12 (1998) 111-177;   [4] M. Bauer, O. Golinelli, Journal of Integer Sequences, Vol 3, (2000). 
E. Caglioti