In the mathematical area of graph theory, a cage is a regular graph that has as few vertices as possible for its girth. Formally, an -graph is defined to be a graph in which each vertex has exactly r neighbors, and in which the shortest cycle has length exactly g. It is known that an -graph exists for any combination of r ≥ 2 and g ≥ 3. An -cage is an -graph with the fewest possible number of vertices, among all -graphs. If a Moore graph exists with degreer and girth g, it must be a cage. Moreover, the bounds on the sizes of Moore graphs generalize to cages: any cage with odd girthg must have at least vertices, and any cage with even girth g must have at least vertices. Any -graph 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 r and g. For instance there are three nonisomorphic -cages, each with 70 vertices : the Balaban 10-cage, the Harries graph and the Harries–Wong graph. But there is only one -cage : the Balaban 11-cage.
Known cages
A degree-one graph has no cycle, and a connected degree-two graph has girth equal to its number of vertices, so cages are only of interest for r ≥ 3. The -cage is a complete graphKr+1 on r+1 vertices, and the -cage is a complete bipartite graphKr,r on 2r vertices. Other notable cages include the Moore graphs:
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 Ramanujan graphs 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.
