# 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
- (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:

g r | 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 (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 Ramanujan graphs (Lubotzky, Phillips & Sarnak 1988) satisfy the bound

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.

## References

- Biggs, Norman (1993),
*Algebraic Graph Theory*(2nd ed.), Cambridge Mathematical Library, pp. 180–190, ISBN 0-521-45897-8. - Bollobás, Béla; Szemerédi, Endre (2002), "Girth of sparse graphs",
*Journal of Graph Theory*,**39**(3): 194–200, doi:10.1002/jgt.10023, MR 1883596. - Exoo, G; Jajcay, R (2008), "Dynamic Cage Survey", Dynamic Surveys,
*Electronic Journal of Combinatorics*,**DS16**, archived from the original on 2015-01-01, retrieved 2012-03-25. - Erdős, Paul; Rényi, Alfréd; Sós, Vera T. (1966), "On a problem of graph theory" (PDF),
*Studia Sci. Math. Hungar.*,**1**: 215–235, archived from the original (PDF) on 2016-03-09, retrieved 2010-02-23. - Hartsfield, Nora; Ringel, Gerhard (1990),
*Pearls in Graph Theory: A Comprehensive Introduction*, Academic Press, pp. 77–81, ISBN 0-12-328552-6. - Holton, D. A.; Sheehan, J. (1993),
*The Petersen Graph*, Cambridge University Press, pp. 183–213, ISBN 0-521-43594-3. - Lazebnik, F.; Ustimenko, V. A.; Woldar, A. J. (1995), "A new series of dense graphs of high girth",
*Bulletin of the American Mathematical Society*, New Series,**32**(1): 73–79, arXiv:math/9501231, doi:10.1090/S0273-0979-1995-00569-0, MR 1284775. - Lubotzky, A.; Phillips, R.; Sarnak, P. (1988), "Ramanujan graphs",
*Combinatorica*,**8**(3): 261–277, doi:10.1007/BF02126799, MR 0963118. - Tutte, W. T. (1947), "A family of cubical graphs",
*Proc. Cambridge Philos. Soc.*,**43**(4): 459–474, Bibcode:1947PCPS...43..459T, doi:10.1017/S0305004100023720.

## External links

- Brouwer, Andries E. Cages
- Royle, Gordon. Cubic Cages and Higher valency cages
- Weisstein, Eric W. "Cage Graph".
*MathWorld*.