On the distribution of the number of special kind chains in a marked complete graph

In this paper we consider the distribution of the number of chains of identical labels of vertices in a complete graph where labels are assigned to vertices randomly according to a given distribution on a finite set and independently of each other. The central limit theorem is proved for the number of such chains when the number of vertices tends to infinity and the chain length remains fixed, including the series scheme (when probabilities of labels assigned to vertices can change with increasing number of vertices of the graph). For a part of parameter variation area we built an estimation of distance between distribution function of number of chains of specified kind and distribution function of standard normal law in uniform metrics. Using numerical simulation it was determined that normal approximation can be applied to the distribution of number of chains of vertex labels on complete graphs with number of vertices of the order of hundred.