On Subgraphs of Bounded Degeneracy in Hypergraphs

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9941)

Abstract

A k-uniform hypergraph has degeneracy bounded by d if every induced subgraph has a vertex of degree at most d. Given a k-uniform hypergraph \(H=(V(H), E(H))\), we show there exists an induced subgraph of size at least
$$\begin{aligned} \sum _{v \in V(H)} \min \left\{ 1, \, c_{k}\, \left( \frac{d+1}{d_{H} (v)+1}\right) ^{1/(k-1)} \right\} , \end{aligned}$$
where \(c_{k} = 2^{- \left( 1 + \frac{1}{k-1} \right) }\left( 1-\frac{1}{k}\right) \) and \(d_{H}(v)\) denotes the degree of vertex v in the hypergraph H. This extends and generalizes a result of Alon-Kahn-Seymour (Graphs and Combinatorics, 1987) for graphs, as well as a result of Dutta-Mubayi-Subramanian (SIAM Journal on Discrete Mathematics, 2012) for linear hypergraphs, to general k-uniform hypergraphs. We also generalize the results of Srinivasan and Shachnai (SIAM Journal on Discrete Mathematics, 2004) from independent sets (0-degenerate subgraphs) to d-degenerate subgraphs. We further give a simple non-probabilistic proof of the Dutta-Mubayi-Subramanian bound for linear k-uniform hypergraphs, which extends the Alon-Kahn-Seymour (Graphs and Combinatorics, 1987) proof technique to hypergraphs. Our proof combines the random permutation technique of Bopanna-Caro-Wei (see e.g. The Probabilistic Method, N. Alon and J. H. Spencer; Dutta-Mubayi-Subramanian) and also Beame-Luby (SODA, 1990) together with a new local density argument which may be of independent interest. We also provide some applications in discrete geometry, and address some natural algorithmic questions.

Keywords

Degenerate graphs Independent sets Hypergraphs Random permutations 

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Copyright information

© Springer-Verlag GmbH Germany 2016

Authors and Affiliations

  1. 1.DataShape, Inria Sophia Antipolis – MéditerranéeMéditerranéeFrance
  2. 2.ACM Unit, Indian Statistical InstituteKolkataIndia

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