The Degree Diameter Problem for Bipartite Graphs

Introduction
The degree/diameter problem for bipartite graphs can be stated as follows:

Given natural numbers d and k, find the largest possible number Nb(d,k) of vertices in a bipartite graph of maximum degree d and diameter k.

An upper bound for Nb(d,k) is given by the so-called bipartite Moore bound Mb(d,k)=2((d-1)k-2)(d-2)-1. Bipartite (d,k)-graphs whose order attains the bipartite Moore bound are called bipartite Moore graphs.

Bipartite Moore graphs have proved to be very rare. Feit and Higman, and also independently Singleton, proved that such graphs exist only when the diameter is 2,3,4 or 6. In the cases when the diameter is 3, 4 or 6, they have been constructed only when d-1 is a prime power.

Therefore, in attempting to settle the values of Nb(d,k), research activities in this problem have follow the following two directions:


 * Increasing the lower bounds for Nb(d,k) by constructing ever larger graphs.

whose order is close to the bipartite Moore bounds Mb(d,k)=2((d-1)k-1)(d-2)-1.
 * Lowering and/or setting upper bounds for Nb(d,k) by proving the non-existence of graphs

Increasing the lower bounds for Nb(d,k)
In recent years there has not been much activity in the constructions of large bipartite graphs. This may be, in part, because there was not an online table showing the latest constructions. In this direction Charles Delorme (in some cases collaborating with Bond and G&oacute;mez-Mart&iacute;) provided some large bipartite graphs by using graph compounding, the concept of partial Cayley graph, and other techniques.

Now, with the release of this online table (see below), we expect to stimulate further research on this area.

Below is the table of the largest known bipartite graphs (as of January 2012) in the undirected degree diameter problem for bipartite graphs of degree at most 3 ≤ d ≤ 16 and diameter 3 ≤ k ≤ 10. This table represents the best lower bounds known at present on the order of (d,k)-bipartite graphs. Many of the graphs of diameter 3 ,4 and 6 are bipartite Moore graphs, and thus are optimal. All optimal graphs are marked in bold.

Table of the orders of the largest known bipartite graphs
The following table is the key to the colors in the table presented above:

Lowering and/or setting upper bounds for Nb(d,k)
The Moore bound can be reached in some cases, but not always in general. Some theoretical work was done to determine the lowest upper bounds. In this direction reserachers have been interested in bipartite graphs of maximum degree d, diameter k and order Mb(d,k)-&delta; for small &delta;. The parameter &delta; is called the defect. Such graphs are called bipartite (d,k,-&delta;)-graphs.

The bipartite (d,k,-2;)-graphs constitute the first interesting family of graphs to be studied. When d&ge;3 and k=2, bipartite (d,k,-2)-graphs are the complete bipartite graphs with partite sets of orders p and q, where either p=q=d-1 or p=d and q=d-2. For d&ge;3 and k&ge;3 only two such graphs are known; a unique bipartite (3, 3,-2)-graph and a unique bipartite (4, 3,-2)-graph.

Studies on bipartite (d,k,-2;)-graphs have been carried out by Charles Delorme, Leif Jorgensen, Mirka Miller and Guillermo Pineda-Villavicencio. They proved several necessary conditions for the existence of bipartite (d,3,-2;)-graphs, the uniqueness of the two known bipartite (d,k,-2;)-graphs for d&ge;3 and k&ge;3, and the non-existence of bipartite (d,k,-2;)-graphs for d&ge;3 and k&ge;4.

Lowest known upper bounds and the percentage of the order of the largest known bipartite graphs
The following table is the key to the colors in the table presented above: