|Does a Moore graph with girth 5 and degree 57 exist?|
An equivalent definition of a Moore graph is that it is a graph of diameter k with girth 2k + 1. Moore graphs were named by Hoffman & Singleton (1960) after Edward F. Moore, who posed the question of describing and classifying these graphs.
As well as having the maximum possible number of vertices for a given combination of degree and diameter, Moore graphs have the minimum possible number of vertices for a regular graph with given degree and girth. That is, any Moore graph is a cage (Erdős, Rényi & Sós 1966). The formula for the number of vertices in a Moore graph can be generalized to allow a definition of Moore graphs with even girth as well as odd girth, and again these graphs are cages.
Bounding vertices by degree and diameter
Let G be any graph with maximum degree d and diameter k, and consider the tree formed by breadth first search starting from any vertex v. This tree has 1 vertex at level 0 (v itself), and at most d vertices at level 1 (the neighbors of v). In the next level, there are at most d(d-1) vertices: each neighbor of v uses one of its adjacencies to connect to v and so can have at most d-1 neighbors at level 2. In general, a similar argument shows that at any level 1 ≤ i ≤ k, there can be at most d(d-1)i vertices. Thus, the total number of vertices can be at most
Hoffman & Singleton (1960) originally defined a Moore graph as a graph for which this bound on the number of vertices is met exactly. Therefore, any Moore graph has the maximum number of vertices possible among all graphs with maximum degree d and diameter k.
Later, Singleton (1968) showed that Moore graphs can equivalently be defined as having diameter k and girth 2k+1; these two requirements combine to force the graph to be d-regular for some d and to satisfy the vertex-counting formula.
Moore graphs as cages
Instead of upper bounding the number of vertices in a graph in terms of its maximum degree and its diameter, we can calculate via similar methods a lower bound on the number of vertices in terms of its minimum degree and its girth (Erdős, Rényi & Sós 1966). Suppose G has minimum degree d and girth 2k+1. Choose arbitrarily a starting vertex v, and as before consider the breadth-first search tree rooted at v. This tree must have one vertex at level 0 (v itself), and at least d vertices at level 1. At level 2 (for k > 1), there must be at least d(d-1) vertices, because each vertex at level 1 has at least d-1 remaining adjacencies to fill, and no two vertices at level 1 can be adjacent to each other or to a shared vertex at level 2 because that would create a cycle shorter than the assumed girth. In general, a similar argument shows that at any level 1 ≤ i ≤ k, there must be at least d(d-1)i vertices. Thus, the total number of vertices must be at least
In a Moore graph, this bound on the number of vertices is met exactly. Each Moore graph has girth exactly 2k+1: it does not have enough vertices to have higher girth, and a shorter cycle would cause there to be too few vertices in the first k levels of some breadth first search tree. Therefore, any Moore graph has the minimum number of vertices possible among all graphs with minimum degree d and diameter k: it is a cage.
For even girth 2k, one can similarly form a breadth-first search tree starting from the midpoint of a single edge. The resulting bound on the minimum number of vertices in a graph of this girth with minimum degree d is
(The right hand side of the formula instead counts the number of vertices in a breadth first search tree starting from a single vertex, accounting for the possibility that a vertex in the last level of the tree may be adjacent to d vertices in the previous level.) Thus, the Moore graphs are sometimes defined as including the graphs that exactly meet this bound. Again, any such graph must be a cage.
The Hoffman–Singleton theorem states that any Moore graph with girth 5 must have degree 2, 3, 7, or 57. The Moore graphs are:
- The complete graphs on n > 2 nodes. (diameter 1, girth 3, degree n-1, order n)
- The odd cycles . (diameter n, girth 2n+1, degree 2, order 2n+1)
- The Petersen graph. (diameter 2, girth 5, degree 3, order 10)
- The Hoffman–Singleton graph. (diameter 2, girth 5, degree 7, order 50)
- A hypothetical graph of diameter 2, girth 5, degree 57 and order 3250; it is currently unknown whether such a graph exists.
If the generalized definition of Moore graphs that allows even girth graphs is used, the Moore graphs also include the even cycles , the complete bipartite graphs with girth four, the Heawood graph with degree 3 and girth 6, and the Tutte–Coxeter graph with degree 3 and girth 8. More generally, it is known (Bannai & Ito 1973; Damerell 1973) that, other than the graphs listed above, all Moore graphs must have girth 5, 6, 8, or 12.
- Bollobás, Béla (1998), "Chap.VIII.3", Modern graph theory, Graduate Texts in Mathematics 184, Springer-Verlag.
- Bannai, E.; Ito, T. (1973), On finite Moore graphs, Journal of the Faculty of Science, the University of Tokyo. Sect. 1 A, Mathematics 20: 191–208, MR 0323615
- Damerell, R. M. (1973), On Moore graphs, Mathematical Proceedings of the Cambridge Philosophical Society 74: 227–236, doi:10.1017/S0305004100048015, MR 0318004
- Erdős, Paul; Rényi, Alfréd; Sós, Vera T. (1966), On a problem of graph theory, Studia Sci. Math. Hungar. 1: 215–235.
- Hoffman, Alan J.; Singleton, Robert R. (1960), Moore graphs with diameter 2 and 3, IBM Journal of Research and Development 5 (4): 497–504, doi:10.1147/rd.45.0497, MR 0140437
- Singleton, Robert R. (1968), There is no irregular Moore graph, American Mathematical Monthly 75 (1): 42–43, doi:10.2307/2315106, MR 0225679