On bipartite graphs of defect at most 4

TL;DR

This paper investigates bipartite graphs close to the Moore bipartite bound with defect at most 4, providing classifications, necessary conditions, and non-existence results for various degrees and diameters.

Contribution

It offers a comprehensive catalogue of bipartite graphs with small defect, establishes necessary conditions, and proves non-existence for certain parameters, advancing understanding of near-optimal bipartite graphs.

Findings

Complete catalogue of bipartite (3,D,-ε)-graphs for 0 ≤ ε ≤ 4

Complete catalogue of bipartite (Δ,D,-ε)-graphs for Δ ≥ 2, 5 ≤ D ≤ 187, 0 ≤ ε ≤ 4

Non-existence of bipartite (Δ,D,-4)-graphs for Δ ≥ 3 and odd D ≥ 7

Abstract

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