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

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Abstract

We consider graphs of maximum degree 3, diameter D2 and at most 4 vertices less than the Moore bound M3,D, that is, (3,D,ϵ)-graphs for ϵ4.

We prove the non-existence of (3,D,4)-graphs for D5, completing in this way the catalogue of (3,D,ϵ)-graphs with D2 and ϵ4. 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,DM3,D6 for D5.

Keywords

Degree/diameter problem
Cubic graphs
Moore bound
Moore graphs
Defect

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