# Cage (graph theory)

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 (*r*,*g*)-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 (*r*,*g*)-graph exists for any combination of *r* ≥ 2 and *g* ≥ 3. An (*r*,*g*)-cage is an (*r*,*g*)-graph with the fewest possible number of vertices, among all (*r*,*g*)-graphs.

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

vertices, and any cage with even girth *g* must have at least

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 nonisomorphic (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 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 (*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 2*r* vertices.

Other notable cages include the Moore graphs:

- (3,5)-cage: the Petersen graph, 10 vertices
- (3,6)-cage: the Heawood graph, 14 vertices
- (3,8)-cage: the Tutte–Coxeter graph, 30 vertices
- (3,10)-cage: the Balaban 10-cage, 70 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:

g: | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

r = 3: |
4 | 6 | 10 | 14 | 24 | 30 | 58 | 70 | 112 | 126 |
---|---|---|---|---|---|---|---|---|---|---|

r = 4: |
5 | 8 | 19 | 26 | 67 | 80 | 728 | |||

r = 5: |
6 | 10 | 30 | 42 | 170 | 2730 | ||||

r = 6: |
7 | 12 | 40 | 62 | 312 | 7812 | ||||

r = 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 Template:Harv. 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 Ramanujan graphs Template:Harv satisfy the bound

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.

## References

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## External links

- Royle, Gordon. Cubic Cages and Higher valency cages