On the path cover number of k-assignable arbitrarily partitionable graphs

Julien Bensmail (1)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique

A graph G is arbitrarily partitionable if for every partition pi = (n_1, n_2, ..., n_p) of |V(G)| there is a partition (V_1, V_2, ..., V_p) of V(G) such that G[V_i] is a connected graph on n_i vertices for every i in {1, 2, ..., p}. If additionally any k arbitrary vertices of G can each be assigned to one part of the resulting vertex partition, then G is a k-assignable arbitrarily partitionable graph. All k-assignable arbitrarily partitionable graphs exhibited so far have an Hamiltonian path. Using the notion of path cover, we show that this Hamiltonian condition is not a necessary one, in the sense that k-assignable arbitrarily partitionable graphs can have arbitrarily small longest paths (compared to their orders).

Domaines: Mathématique discrète [cs.DM]

Julien Bensmail. On the path cover number of k-assignable arbitrarily partitionable graphs. 2013. ⟨hal-00881861⟩