Partitioning Harary graphs into connected subgraphs containing prescribed vertices

A graph G is arbitrarily partitionable (AP for short) if for every partition (n1, n2, ..., np) of |V (G)| there exists a partition (V1, V2, ..., Vp) of V (G) such that each Vi induces a connected subgraph of G with order ni. If, additionally, k of these subgraphs (k ≤ p) each contains an arbitrary vertex of G prescribed beforehand, then G is arbitrarily partitionable under k prescriptions (AP+k for short). Every AP+k graph on n vertices is (k + 1)-connected, and thus has at least d 2 e edges. We show that there exist AP+k graphs on n vertices and d 2 e edges for every k ≥ 1 and n ≥ k.