Cage (graph theory)

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 Harries–Wong graph. But there is only one : the .

Known cages

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 ≥ 3. The -cage is a complete graph K_r+1 on r + 1 vertices, and the -cage is a complete bipartite graph K_r,''r on 2r'' vertices.
Notable cages include:

-cage: the Petersen graph, 10 vertices
-cage: the Heawood graph, 14 vertices
-cage: the McGee graph, 24 vertices
-cage: the Tutte–Coxeter graph, 30 vertices
-cage: the Balaban 10-cage, 70 vertices
-cage: the Balaban 11-cage, 112 vertices
-cage: the Tutte 12-cage, 126 vertices
-cage: the Robertson graph, 19 vertices
-cage: The Hoffman–Singleton graph, 50 vertices.
When r − 1 is a prime power, the cages are the incidence graphs of projective planes.
When r − 1 is a prime power, the and cages are generalized polygons.

The numbers of vertices in the known cages, for values of r > 2 and g > 2, other than projective planes and generalized polygons, are:

	3	4	5	6	7	8	9	10	11	12
3	4	6	10	14	24	30	58	70	112	126
4	5	8	19	26	67	80				728
5	6	10	30	42		170				2730
6	7	12	40	62		312				7812
7	8	14	50	90

Asymptotics

For large values of g, the Moore bound implies that the number n of vertices must grow at least singly 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. 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.