Lower connectivities of regular graphs with small diameter

Krishnamoorthy, Thulasiraman and Swamy [Minimum order graphs with specified diameter, connectivity and regularity, Networks 19 (1989) 25–46] showed that a δ-regular graph with diameter D at most 3 has (vertex-)connectivity κ at least 2, and if D ≤ 2 then the connectivity is at least κ ≥ min{δ, 3}. Likewise, Soneoka, Nakada, Imase and Peyrat [Sufficient conditions for maximally connected graphs, Discrete Mathematics 63 (1) (1987) 53–66] proved that a graph with diameter D ≤ 2b(g − 1)/2c − 1 has maximum connectivity κ = δ. In this work we generalize and improve these results for δ-regular graphs. More precisely we prove that if D ≤ 2b(g − 1)/2c + 1 then κ ≥ 2, and if D ≤ g − 1 then κ ≥ min{δ, 3}. Furthermore, we prove for g even that if D ≤ g − 2 then κ ≥ min{δ, 6}, and for bipartite δ-regular graphs we obtain that if D ≤ g−1 then κ ≥ min{δ, 4}, and if D ≤ g then κ ≥ 2. We establish similar bounds for the edge connectivity and present some examples showing that these results are best possible, at least for particular values of the girth and the regularity δ.