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 a length of 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
vertices, and any cage with even girth must have at least
vertices. Any with exactly this many vertices is by definition a Moore graph and therefore automatically a cage.
There may exist multiple cages for a given combination of and . For instance there are three non-isomorphic , each with 70 vertices: the , the Harries graph and the HarriesâÂÂWong graph. But there is only one : the (with 112 vertices).
A 1-regular graph has no cycle, and a connected 2-regular graph has girth equal to its number of vertices, so cages are only of interest for r âÂÂ¥ 3. The (r,3)-cage is a complete graph K<sub>r+1</sub> on r + 1 vertices, and the (r,4)-cage is a complete bipartite graph K<sub>r,r</sub> on 2r vertices.
Notable cages include:
The numbers of vertices in the known (r,g) cages, for values of r > 2 and g > 2, other than projective planes and generalized polygons, are:
For large values of g, the Moore bound implies that the number n of vertices must grow at least exponentially as a function of g. Equivalently, g can be at most proportional to the logarithm of n. More precisely,
It is believed that this bound is tight or close to tight . The best known lower bounds on g are also logarithmic, but with a smaller constant factor (implying that n grows exponentially but at a higher rate than the Moore bound). Specifically, the construction of Ramanujan graphs defined by satisfy the bound
This bound was improved slightly by .
It is unlikely that these graphs are themselves cages, but their existence gives an upper bound to the number of vertices needed in a cage.