On Subgraphs of Bounded Degeneracy in Hypergraphs

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 Sigma(v is an element of 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 vertex v in the hypergraph H. This extends and generalizes a result of AlonKahn-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.