In this chapter, we consider an edge coloring problem in graphs that can be used to model certain situations, one of which we now describe. Suppose that a hard-line network of n computers is to be constructed. Each of these computers requires k different types of connections. There are r locations on the back of each computer at which ports can be placed. Each computer needs to have at least one connection of each type and, for security reasons, no two computers can have more than one connection between them. In order to maximize the number of fail-safe connections, every port is to be used. Furthermore, it is advantageous for a computer technician to be able to distinguish the computers based only on the number of types of connections they have. For which values of n, k and r is such a situation possible?