Some properties of minimal arbitrarily partitionable graphs

A graph G on n vertices is arbitrarily partitionable (AP for short) if for every partition (?(1), . . . ,?(p)) of n (that is, ?(1) + . . . + ?(p) = n), the vertex set V(G) can be partitioned into p parts V-1, ... , V-p such that G[V-i] has order ?(i) and is connected for every i ? {1, ... , p}. We investigate minimal AP graphs, which are those AP graphs that are not spanned by any proper AP subgraph. We pursue previous investigations by Ravaux and Baudon, Przybylo, and Wozniak, who established that minimal AP graphs are not all trees, but conjectured that they should all be somewhat sparse. We investigate several aspects of minimal AP graphs, including their minimum degree, their maximum degree, and their clique number. Some of the results we establish arise from an exhaustive list we give, of all minimal AP graphs of order at most 10. We also address new questions on the structure of minimal AP graphs.