On the longest path in a recursively partitionable graph

Julien Bensmail (1)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique

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 K_1, or for every sequence (n_1, ..., n_p) of positive integers summing up to n there exists a partition (V_1, ..., V_p) of V(G) such that each V_i induces a connected R-AP subgraph of G on n_i 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 also holds for R-AP graphs with arbitrary connectivity.

Mots clés: recursively partitionable graph; longest elementary path

Domaines: Mathématique discrète [cs.DM]

Julien Bensmail. On the longest path in a recursively partitionable graph. Opuscula Mathematica, 2013, 33 (4), http://www.opuscula.agh.edu.pl. ⟨hal-00718029v4⟩