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 $r \geq 2$ 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