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 $$H=VH, EH$$ , we show there exists an induced subgraph of size at least $$\begin{aligned} \sum _{v \in VH} \min \left\{ 1, \, c_{k}\, \left \frac{d+1}{d_{H} v+1}\right ^{1/k-1} \right\} , \end{aligned}$$ 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.