Partitioning harary graphs into connected subgraphs containing prescribed vertices

Abstract

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 ⌈n(k+1)/2⌉ edges. We show that there exist AP+k graphs on n vertices and ⌈n(k+1)/2⌉ edges for every k ≥ 1 and n ≥ k .