# 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 (r, g)-graph is defined to be a graph in which each vertex has exactly r neighbors, and in which the shortest cycle has a length of exactly g. An (r, g)-cage is an (r, g)-graph with the smallest possible number of vertices, among all (r, g)-graphs. A (3, g)-cage is often called a g-cage.
It is known that an (r, g)-graph exists for any combination of r ≥ 2 and g ≥ 3. It follows that all (r, g)-cages exist.
If a Moore graph exists with degree r and girth g, it must be a cage. Moreover, the bounds on the sizes of Moore graphs generalize to cages: any cage with odd girth g must have at least
    $1+r\sum _{i=0}^{(g-3)/2}(r-1)^{i}$
vertices, and any cage with even girth g must have at least
    $2\sum _{i=0}^{(g-2)/2}(r-1)^{i}$
vertices. Any (r, g)-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 non-isomorphic (3, 10)-cages, each with 70 vertices: the Balaban 10-cage, the Harries graph and the Harries–Wong graph. But there is only one (3, 11)-cage: the Balaban 11-cage (with 112 vertices).

## 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 (r,3)-cage is a complete graph K$_{r+1}$ on r + 1 vertices, and the (r,4)-cage is a complete bipartite graph K$_{r,r}$ on 2r vertices.
Notable cages include:
* (3,5)-cage: the Petersen graph, 10 vertices
* (3,6)-cage: the Heawood graph, 14 vertices
* (3,7)-cage: the McGee graph, 24 vertices
* (3,8)-cage: the Tutte–Coxeter graph, 30 vertices
* (3,10)-cage: the Balaban 10-cage, 70 vertices
* (3,11)-cage: the Balaban 11-cage, 112 vertices
* (4,5)-cage: the Robertson graph, 19 vertices
* (7,5)-cage: The Hoffman–Singleton graph, 50 vertices.
* When r − 1 is a prime power, the (r,6) cages are the incidence graphs of projective planes.
* When r − 1 is a prime power, the (r,8) and (r,12) cages are generalized polygons.
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:
| gr | 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,
    $g\leq 2\log _{r-1}n+O(1).$
It is believed that this bound is tight or close to tight (Bollobás & Szemerédi 2002). The best known lower bounds on g are also logarithmic, but with a smaller constant factor (implying that n grows singly exponentially but at a higher rate than the Moore bound). Specifically, the construction of Ramanujan graphs defined by Lubotzky, Phillips & Sarnak (1988) satisfy the bound
    $g\geq {\frac {4}{3}}\log _{r-1}n+O(1).$
This bound was improved slightly by Lazebnik, Ustimenko & Woldar (1995).
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.