0 On bipartite graphs of defect at most 4 Ramiro

We consider the bipartite version of the degree/diameter problem, namely, given natural numbers ∆ ≥ 2 and D ≥ 2, find the maximum number Nb(∆,D) of vertices in a bipartite graph of maximum degree ∆ and diameter D. In this context, the Moore bipartite bound Mb(∆,D) represents an upper bound for Nb(∆,D). Bipartite graphs of maximum degree ∆, diameter D and order Mb(∆,D), called Moore bipartite graphs, turned out to be very rare. Therefore, it is very interesting to investigate bipartite graphs of maximum degree ∆ ≥ 2, diameter D ≥ 2 and order Mb(∆,D) − ǫ with small ǫ > 0, that is, bipartite (∆,D,−ǫ)-graphs. The parameter ǫ is called the defect. This paper considers bipartite graphs of defect at most 4, and presents all the known such graphs. Bipartite graphs of defect 2 have been studied in the past; if ∆ ≥ 3 and D ≥ 3, they may only exist for D = 3. However, when ǫ > 2 bipartite (∆,D,−ǫ)-graphs represent a wide unexplored area. The main results of the paper include the complete catalogue of bipartite (3,D,−ǫ)-graphs with D ≥ 2 and 0 ≤ ǫ ≤ 4; the complete catalogue of bipartite (∆,D,−ǫ)-graphs with ∆ ≥ 2, [email protected] Corresponding author: [email protected] 1 5 ≤ D ≤ 187 (D 6= 6) and 0 ≤ ǫ ≤ 4; and a non-existence proof of all bipartite (∆,D,−4)-graphs with ∆ ≥ 3 and odd D ≥ 7. Finally, we conjecture that there are no bipartite graphs of defect 4 for ∆ ≥ 3 and D ≥ 5, and comment on some implications of our results for upper bounds of Nb(∆,D).