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Cage (graph theory)

Regular graph with fewest possible nodes for its girth

Wikipedia contributorsAdapted for Fathomly · View authors
3 min read
Source updated February 24, 2026
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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 ≥ 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

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).

01Known 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:

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 3456789101112
3 46101424305870112126
4 5819266780728
5 61030421702730
6 71240623127812
7 8145090

02Asymptotics

For large values of g, the Moore bound implies that the number n of vertices must grow at least 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 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.

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Sources and credits

This article is adapted from the Wikipedia article “Cage (graph theory)”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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