On minimal arbitrarily partitionable graphs

Authors:
Olivier Baudon;Jakub PrzybyłO;Mariusz Woniak
Affiliations:
Univ. Bordeaux, LaBRI, UMR 5800, 33400 Talence, France and CNRS, LaBRI, UMR 5800, 33400 Talence, France;AGH University of Science and Technology, al. A. Mickiewicza 30, 30-059 Krakow, Poland;AGH University of Science and Technology, al. A. Mickiewicza 30, 30-059 Krakow, Poland
Venue:
Information Processing Letters
Year:
2012

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Abstract

A graph G=(V,E) of order n is called arbitrarily partitionable, or AP for short, if given any sequence of positive integers n"1,...,n"k summing up to n, we can always partition V into subsets V"1,...,V"k of sizes n"1,...,n"k, 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|==1+130 (if such C exists).