On the longest path in a recursively partitionable graph

A connected graph G with order n ≥ 1 is said to be recursively arbitrarily partitionable (R-AP for short) if either it is isomorphic to K1, or for every sequence (n1, . . . , np) of positive integers summing up to n there exists a partition (V1, . . . , Vp) of V (G) such that each Vi induces a connected R-AP subgraph of G on ni vertices. Since previous investigations, it is believed that a R-AP graph should be “almost traceable” somehow. We first show that the longest path of a R-AP graph on n vertices is not constantly lower than n for every n. This is done by exhibiting a graph family C such that, for every positive constant c ≥ 1, there is a R-AP graph in C that has arbitrary order n and whose longest path has order n−c. We then investigate the largest positive constant c’ < 1 such that every R-AP graph on n vertices has its longest path passing through n • c’ vertices. In particular, we show that c’ ≥ 2/3 . This result holds for R-AP graphs with arbitrary connectivity.