The minimal vertex extensions for colored complete graphs

The article proposes the results of the search for minimal vertex extensions of undirected colored complete graphs. The research topic is related to the modelling of full fault tolerant technical systems with a different type of their objects in the terminology of graph theory. Let a technical system be $\Sigma$, then there is a graph $G(\Sigma)$, which vertices reflects system's objects and edges reflects connections between these objects. Type of each object reflected in a mapping of some color from $F = \{1,2\dots,i\}$ to the corresponding vertex. System's $\Sigma$ vertex extension is a graph $G(\Sigma)$ which contains additional vertices. System reflected by graph $G(\Sigma)$ can work even if there are $k$ faults of its objects. Complete graph is a graph where each two vertices have an edge between them. Complete graphs have no edge extensions because there is no way to add additional edge to the graph with a maximum number of edges. In other words, the system reflected by some complete graph cannot be able to resist connection faults. Therefore the article research is focused on vertex extensions only. There is a description of vertex extensions existence condition for those colored complete graphs. This paper considers generating schemes for such minimal vertex extensions along with formulas, which allows to calculate number of additional edges to have an ability to construct minimal vertex extension.