On bipartite graphs of defect at most 4

We consider the bipartite version of the degree/diameter problem, namely, given natural numbers Δ \geq 2 and D \geq 2, find the maximum number Nb(Δ,D) of vertices in a bipartite graph of maximum degree Δ and diameter D. In this context, the Moore bipartite bound Mb(Δ,D) represents an upper bound for Nb(Δ,D). Bipartite graphs of maximum degree Δ, diameter D and order Mb(Δ,D), called Moore bipartite graphs, have turned out to be very rare. Therefore, it is very interesting to investigate bipartite graphs of maximum degree Δ \geq 2, diameter D \geq 2 and order Mb(Δ,D) - εwith small ε> 0, that is, bipartite (Δ,D,-ε)-graphs. The parameter εis 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 Δ \geq 3 and D \geq 3, they may only exist for D = 3. However, when ε> 2 bipartite (Δ,D,-ε)-graphs represent a wide unexplored area. The main results of the paper include several necessary conditions for the existence of bipartite $(Δ,d,-4)$-graphs; the complete catalogue of bipartite (3,D,-ε)-graphs with D \geq 2 and 0 \leq ε\leq 4; the complete catalogue of bipartite (Δ,D,-ε)-graphs with Δ \geq 2, 5 \leq D \leq 187 (D /= 6) and 0 \leq ε\leq 4; and a non-existence proof of all bipartite (Δ,D,-4)-graphs with Δ \geq 3 and odd D \geq 7. Finally, we conjecture that there are no bipartite graphs of defect 4 for Δ \geq 3 and D \geq 5, and comment on some implications of our results for upper bounds of Nb(Δ,D).

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