On Subgraphs of Bounded Degeneracy in Hypergraphs

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.