cage

In the mathematical field of graph theory, a cage is a regular graph that has as few vertices as possible for its girth. Formally, an is defined to be a graph in which each vertex has exactly neighbors, and in which the shortest cycle has length exactly . An is an with the smallest possible number of vertices, among all . A is often called a . It is known that an exists for any combination of and . It follows that all exist. If a Moore graph exists with degree and girth , it must be a cage. Moreover, the bounds on the sizes of Moore graphs generalize to cages: any cage with odd girth must have at least