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

We consider graphs of maximum degree 3, diameter D≥2D≥2 and at most 4 vertices less than the Moore bound M3,DM3,D, that is, (3,D,−ϵ)(3,D,−ϵ)-graphs for ϵ≤4ϵ≤4.We prove the non-existence of (3,D,−4)(3,D,−4)-graphs for D≥5D≥5, completing in this way the catalogue of (3,D,−ϵ)(3,D,−ϵ)-graphs with D≥2D≥2 and ϵ≤4ϵ≤4. Our results also give an improvement to the upper bound on the largest possible number N3,DN3,D of vertices in a graph of maximum degree 3 and diameter DD, so that N3,D≤M3,D−6N3,D≤M3,D−6 for D≥5D≥5.