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 $(3,8)$-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 $(r, g)$-cage is often called a $g$-cage.

It is known that an $(3, g)$-graph exists for any combination of $(r, g)$ and $r ≥ 2$. It follows that all $g ≥ 3$-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 nonisomorphic $(r, g)$-cages, each with 70 vertices: the Balaban 10-cage, the Harries graph and the Harries–Wong graph. But there is only one $(3, 10)$-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 Kr+1 on r+1 vertices, and the (r,4)-cage is a complete bipartite graph Kr,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,
 * $$g\le 2\log_{r-1} n + O(1).$$

It is believed that this bound is tight or close to tight. 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 satisfy the bound
 * $$g\ge \frac{4}{3}\log_{r-1} n + O(1).$$

This bound was improved slightly by.

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.