On subgraphs of bounded degeneracy in hypergraphs
Document Type
Conference Article
Publication Title
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Abstract
A k-uniform hypergraph has degeneracy bounded by d if every induced subgraph has a vertex of degree at most d. Given a kuniform hypergraph H = (V (H),E(H)), we show there exists an induced subgraph of size at least (formula presented), where ck = 2−(1+ 1/k−1) (1 – 1/k) and dH(υ) denotes the degree of vertex υ 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 nonprobabilistic 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.
First Page
295
Last Page
306
DOI
10.1007/978-3-662-53536-3_25
Publication Date
1-1-2016
Recommended Citation
Dutta, Kunal and Ghosh, Arijit, "On subgraphs of bounded degeneracy in hypergraphs" (2016). Conference Articles. 766.
https://digitalcommons.isical.ac.in/conf-articles/766
