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The gossip problem (telephone problem) is an information disseminationproblem in which each of $n$ nodes of a communication network has a uniquepiece of information that must be transmitted to all the other nodes usingtwo-way communications (telephone calls) between the pairs of nodes. During acall between the given two nodes, they exchange the whole information known tothem at that moment. In this paper we investigate the $k$-fault-tolerant gossipproblem, which is a generalization of the gossip problem, where at most $k$arbitrary faults of calls are allowed. The problem is to find the minimalnumber of calls $\tau(n,k)$ needed to guarantee the $k$-fault-tolerance. Weconstruct two classes of $k$-fault-tolerant gossip schemes (sequences of calls)and found two upper bounds of $\tau(n,k)$, which improve the previously knownresults. 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 boundfor general odd $n$. From the expressions for the second upper bound it followsthat $\tau(n,k) \leq 2/3 n k + O(n)$ for large $n$. Assuming that the calls cantake place simultaneously, it is also of interest to find $k$-fault-tolerantgossip schemes, which can spread the full information in minimal time. For even$n$ we showed that the minimal time is $T(n,k)=\lceil\log_2 n\rceil + k$.

Tighter Upper Bounds for the Minimum Number of Calls and Rigorous Minimal Time in Fault-Tolerant Gossip Schemes