Complete catalogue of graphs of maximum degree 3 and defect at most 4

We consider graphs of maximum degree 3, diameter D ≥ 2 and at most 4 vertices less than the Moore bound M3, D, that is, (3, D, - ε{lunate})-graphs for ε{lunate} ≤ 4. We prove the non-existence of (3, D, - 4)-graphs for D ≥ 5, completing in this way the catalogue of (3, D, - ε{lunate})-graphs with D ≥ 2 and ε{lunate} ≤ 4. Our results also give an improvement to the upper bound on the largest possible number N3, D of vertices in a graph of maximum degree 3 and diameter D, so that N3, D ≤ M3, D - 6 for D ≥ 5.