Suppose we have a group of n people, each possessing an item of information not known to any of the others and that during each unit of time each person can send all of the information he knows to at most k other people. Further suppose that each of at most k other people can send all of the information they know to him. Determine the length of time required, f(n, k), so that all n people know all of rhe information. We show f(n, k) = Pog,+,nl. We define g(n, k) analogously except that no person may both send and receive information during a unit time period. We show rlog,+,nl~g(n, k)s2[ log,+,nl in general and further show that the upper bound can be significantly improveu in the cases k = 1 or 2. We conjecture g(n, k) = b, log L+ln +0(l) for a function bl, we determine. I. Iniroduction The telephone gossip problem is the following: there are n people (gossips) each possessing some information not known by any of the others. They communicate by telephone and, whenever two people talk to one another during a call, each informs the other of all the information known at that time. Determine the minimum number, f(n), of calls needed before all n people know all the information. It is shown in Baker and Shostok (l), Hajnal, Milner and SzemerCdi (5), and Tidjeman (10) that f(1) = 0, f(2) = 1, f(3) = 3 and f(n) = 2n -4 for n 24. This is generalized in Lebensold (8) where it is shown that n people can share all their information by means of f(n, k) k-person party line phone calls where f(n, k)= l(n-2)l(k-1)J + [(n-1)/k] +l, lsnsk', and f(n, k)=2L(n-2)/(k-l)J, n>k*. Harary and Schwenk (6) generalize the problem of determining f(n) to determining f(G,), where G,, denotes a connected graph on n vertices whose t This work was supported by the U.S. Energy Research and Development Administration (ERDA) under Contract No. AT(29-l)-789.