  The gossip problem (telephone problem) is an information dissemination
problem in which each of $n$ nodes of a communication network has a unique
piece of information that must be transmitted to all the other nodes using
two-way communications (telephone calls) between the pairs of nodes. During a
call between the given two nodes, they exchange the whole information known to
them at that moment. In this paper we investigate the $k$-fault-tolerant gossip
problem, which is a generalization of the gossip problem, where at most $k$
arbitrary faults of calls are allowed. The problem is to find the minimal
number of calls $\tau(n,k)$ needed to guarantee the $k$-fault-tolerance. We
construct two classes of $k$-fault-tolerant gossip schemes (sequences of calls)
and found two upper bounds of $\tau(n,k)$, which improve the previously known
results. The first upper bound for general even $n$ is $\tau(n,k) \leq 1/2 n
\lceil\log_2 n\rceil + 1/2 n k$. This result is used to obtain the upper bound
for general odd $n$. From the expressions for the second upper bound it follows
that $\tau(n,k) \leq 2/3 n k + O(n)$ for large $n$. Assuming that the calls can
take place simultaneously, it is also of interest to find $k$-fault-tolerant
gossip schemes, which can spread the full information in minimal time. For even
$n$ we showed that the minimal time is $T(n,k)=\lceil\log_2 n\rceil + k$.
