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 ≥ 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

${\displaystyle 1+r\sum _{i=0}^{(g-3)/2}(r-1)^{i))$

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

${\displaystyle 2\sum _{i=0}^{(g-2)/2}(r-1)^{i))$

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 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 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:

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,

${\displaystyle g\leq 2\log _{r-1}n+O(1).}$

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

${\displaystyle g\geq {\frac {4}{3))\log _{r-1}n+O(1).}$

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.