Elsevier

Discrete Mathematics

Volume 313, Issue 4, 28 February 2013, Pages 381-390
Discrete Mathematics

On large bipartite graphs of diameter 3

https://doi.org/10.1016/j.disc.2012.11.013Get rights and content
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Abstract

We consider the bipartite version of the degree/diameter problem, namely, given natural numbers d2 and D2, find the maximum number Nb(d,D) of vertices in a bipartite graph of maximum degree d and diameter D. In this context, the bipartite Moore bound Mb(d,D) represents a general upper bound for Nb(d,D). Bipartite graphs of order Mb(d,D) are very rare, and determining Nb(d,D) still remains an open problem for most (d,D) pairs.

This paper is a follow-up of our earlier paper (Feria-Purón and Pineda-Villavicencio, 2012 [5]), where a study on bipartite (d,D,4)-graphs (that is, bipartite graphs of order Mb(d,D)4) was carried out. Here we first present some structural properties of bipartite (d,3,4)-graphs, and later prove that there are no bipartite (7,3,4)-graphs. This result implies that the known bipartite (7,3,6)-graph is optimal, and therefore Nb(7,3)=80. 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 (5,3,4)-graph, and the non-existence of bipartite (6,3,4)-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 Nb(11,3).

Keywords

Degree/diameter problem for bipartite graphs
Bipartite Moore bound
Large bipartite graphs
Defect

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