We consider the bipartite version of the degree/diameter problem, namely, given natural numbers and , find the maximum number of vertices in a bipartite graph of maximum degree and diameter . In this context, the bipartite Moore bound represents a general upper bound for . Bipartite graphs of order are very rare, and determining still remains an open problem for most pairs.
This paper is a follow-up of our earlier paper (Feria-Purón and Pineda-Villavicencio, 2012 [5]), where a study on bipartite -graphs (that is, bipartite graphs of order ) was carried out. Here we first present some structural properties of bipartite -graphs, and later prove that there are no bipartite -graphs. This result implies that the known bipartite -graph is optimal, and therefore . We dub this graph the Hafner–Loz graph after its first discoverers Paul Hafner and Eyal Loz.
The approach here presented also provides a proof of the uniqueness of the known bipartite -graph, and the non-existence of bipartite -graphs.
In addition, we discover at least one new largest known bipartite–and also vertex-transitive–graph of degree 11, diameter 3 and order 190, a result which improves by four vertices the previous lower bound for .