On Subgraphs of Bounded Degeneracy in Hypergraphs

Kunal Dutta 1 Arijit Ghosh 2
1 DATASHAPE - Understanding the Shape of Data
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
2 Algorithms and Complexity
MPII - Max-Planck-Institut für Informatik
Abstract : A k-uniform hypergraph is d-degenerate if every induced subgraph has a vertex of degree at most d. Given a k-uniform hyper-graph H = (V (H), E(H)), we show there exists an induced subgraph of size at least $v∈V (H) min (1, c_k d + (1/ d_H (v) + 1)^{1/(k−1)}$ , where $c_k = 2^{−(1+ 1/ k−1)}( 1 − 1/ k)$ and d_H (v) denotes the degree of ver-tex v in the hypergraph H. This connects, extends, and generalizes results of Alon-Kahn-Seymour (1987), on d-degenerate sets of graphs, Dutta-Mubayi-Subramanian (2012) on d-degenerate sets of linear hypergraphs, and Srinivasan-Shachnai (2004) on independent sets in hypergraphs to d-degenerate sub-graphs of hypergraphs. Our technique also gives optimal lower bounds for a more generalised definition of degeneracy introduced by Zaker (2013). We further give a simple non-probabilistic proof of the Dutta-Mubayi-Subramanian bound for linear k-uniform hypergraphs, which extends the Alon, Kahn and Seymour (1987) proof technique to hypergraphs. Finally we provide several applications in discrete geometry, extending results of Payne-Wood (2013) and Cardinal-Tóth-Wood (2016). We also address some natural algorithmic questions. The proof of our main theorem combines the random permutation technique of Bopanna-Caro-Wei and Beame and Luby, together with a new local density argument which may be of independent interest.
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Pré-publication, Document de travail
2017
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Soumis le : jeudi 21 décembre 2017 - 07:26:37
Dernière modification le : jeudi 11 janvier 2018 - 16:38:51

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Kunal Dutta, Arijit Ghosh. On Subgraphs of Bounded Degeneracy in Hypergraphs. 2017. 〈hal-01669886〉

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