Dense Arbitrarily Partitionable Graphs

A graph $G$ of order $n$ is called arbitrarily partitionable (AP for short) if, for every sequence $(n_1, . . ., n_k)$ of positive integers with $n_1 + ⋯ + n_k = n$, there exists a partition $(V_1, . . ., V_k)$ of the vertex set $V(G)$ such that $V_i$ induces a connected subgraph of order $n_i$ for $i = 1, . . ., k$. In this paper we show that every connected graph $G$ of order $n \ge 22$ and with \( ‖G‖ > \binom{n-4}{2} + 12 \) edges is AP or belongs to few classes of exceptional graphs.