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 length 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 $r \geq 2$ and $g \geq 3$ . 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:

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,

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.

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.