Cage (graph theory) explained

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 is defined to be a graph in which each vertex has exactly neighbors, and in which the shortest cycle has length exactly . An is an with the smallest possible number of vertices, among all . A is often called a .

It is known that an exists for any combination of and . It follows that all exist.

If a Moore graph exists with degree and girth, it must be a cage. Moreover, the bounds on the sizes of Moore graphs generalize to cages: any cage with odd girth must have at least

1 + r\sum ^{(g -3 )/ 2} _{i =0} (r -1 ) ^{i} vertices, and any cage with even girth must have at least2 \sum ^{(g -2 )/ 2} _{i =0} (r -1 ) ^{i} vertices. Any 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 and . For instance there are three non-isomorphic, each with 70 vertices: the, the Harries graph and the Harries–Wong graph. But there is only one : the (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:

	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 \le 2log _{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 boundg \ge (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.

References

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External links

• Brouwer, Andries E. Cages
• Royle, Gordon. Cubic Cages and Higher valency cages