On Moore bipartite digraphs

Abstract

In the context of the degree/diameter problem for directed graphs, it is known that the number of vertices of a strongly connected bipartite digraph satisfies a Moore-like bound in terms of its diameter k and the maximum out-degrees (d 1 ; d 2 ) of its partite sets of vertices. It has been proved that, when d 1 d 2 > 1, the digraphs attaining such a bound, called Moore bipartite digraphs, only exist when 2 k 4. This paper deals with the problem of their enumeration. In this context, using the theory of circulant matrices and the so-called De Bruijn near-factorizations of cyclic groups, we present some new constructions of Moore bipartite digraphs of diameter three and composite out-degrees. By applying the iterated line digraph technique, such constructions also provide new families of dense bipartite digraphs with arbitrary diameter. Moreover, we show that the line digraph structure is inherent in any Moore bipartite digraph G of diameter k ¼ 4, which means that G ¼ L G 0 , where G 0 is a Moore bipartite digraph of diameter k ¼ 3. ß

On bipartite‐mixed graphs

“…As for the undirected case, complete bipartite digraphs are the unique bipartite Moore digraphs of diameter , meanwhile for some families of bipartite Moore digraphs have been constructed, although the problem of their enumeration is not closed (see again , and Fiol et al. ).…”

Moore Graphs and Beyond: A survey of the Degree/Diameter Problem

“…is attainable only when k = 2,3 or 4. The interested reader can find out more about bipartite and almost bipartite Moore digraphs in studies by Fiol, Gimbert, Gómez and Wu [161], and Fiol and Gimbert [160]; for multipartite version, see Fiol, Gimbert and Miller [162].…”