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On minimal arbitrarily partitionable graphs

Abstract : 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).
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https://hal.archives-ouvertes.fr/hal-00712852
Contributor : Olivier Baudon Connect in order to contact the contributor
Submitted on : Thursday, June 28, 2012 - 1:42:42 PM
Last modification on : Saturday, June 25, 2022 - 10:33:02 AM

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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⟩

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