On minimal arbitrarily partitionable graphs

Olivier Baudon (1) , Jakub Przybylo (2) , Mariusz Woźniak (2)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique
2 AGH UST - AGH University of Science and Technology [Krakow, PL]

A graph G = (V , E) of order n is called arbitrarily partitionable, or AP for short, if given any sequence of positive integers n1 , . . . , nk summing up to n, we can always partition V into subsets V1,...,Vk of sizes n1,...,nk, resp., inducing connected subgraphs in G. If additionally G is minimal with respect to this property, i.e. it contains no AP spanning subgraph, we call it a minimal AP-graph. It has been conjectured that such graphs are sparse, i.e., there exists an absolute constant C such that |E| ≤ C.n for each of them. We construct a family of minimal AP-graphs which prove that C ≥ 1 + 1/30 (if such C exists).

Mots clés: Graph; Arbitrarily Partitionable; Minimal

Domaines: Mathématique discrète [cs.DM]

Olivier Baudon, Jakub Przybylo, Mariusz Woźniak. On minimal arbitrarily partitionable graphs. Information Processing Letters, 2012, 112, pp.697-700. ⟨10.1016/j.ipl.2012.06.010⟩. ⟨hal-00712852⟩