Bibliographic Details
| Title: |
Edge-statistics on large graphs |
| Authors: |
Alon, Noga, Hefetz, Dan, Krivelevich, Michael, Tyomkyn, Mykhaylo |
| Superior Title: |
Combinatorics, Probability and Computing, 29(2), 163-189, (2020-03) |
| Publisher Information: |
Cambridge University Press |
| Publication Year: |
2020 |
| Collection: |
Caltech Authors (California Institute of Technology) |
| Description: |
The inducibility of a graph H measures the maximum number of induced copies of H a large graph G can have. Generalizing this notion, we study how many induced subgraphs of fixed order k and size ℓ a large graph G on n vertices can have. Clearly, this number is (n k) for every n, k and ℓ ∈ {0, (k 2)}. We conjecture that for every n, k and 0 < ℓ < (k 2) this number is at most 91/e + o_k(1))(n k). If true, this would be tight for ℓ ∈ {1, k − 1}. In support of our 'Edge-statistics Conjecture', we prove that the corresponding density is bounded away from 1 by an absolute constant. Furthermore, for various ranges of the values of ℓ we establish stronger bounds. In particular, we prove that for 'almost all' pairs (k, ℓ) only a polynomially small fraction of the k-subsets of V(G) have exactly ℓ edges, and prove an upper bound of (1/2 + o_k(1)(n k) for ℓ = 1. Our proof methods involve probabilistic tools, such as anti-concentration results relying on fourth moment estimates and Brun's sieve, as well as graph-theoretic and combinatorial arguments such as Zykov's symmetrization, Sperner's theorem and various counting techniques. ; © Cambridge University Press 2019. (Received 17 May 2018; revised 24 January 2019; first published online 14 November 2019) Research supported in part by NSF grant DMS-1855464, ISF grant 281/17, BSF grant 2018267 and the Simons Foundation. The research of Dan Hefetz is supported by ISF grant 822/18. Partially supported by USA-Israel BSF grants 2014361 and 2018267, and by ISF grant 1261/17. The research of Mykhaylo Tyomkyn is supported in part by ERC Starting Grant 633509. We thank the anonymous referee for carefully reading our paper and for providing helpful remarks. ; Submitted - 1805.06848.pdf |
| Document Type: |
article in journal/newspaper |
| Language: |
unknown |
| Relation: |
https://arxiv.org/abs/1805.06848; https://doi.org/10.1017/s0963548319000294; oai:authors.library.caltech.edu:y84ej-bvq89; eprintid:106469; resolverid:CaltechAUTHORS:20201105-160425983 |
| DOI: |
10.1017/s0963548319000294 |
| Availability: |
https://doi.org/10.1017/s0963548319000294 |
| Rights: |
info:eu-repo/semantics/openAccess ; Other |
| Accession Number: |
edsbas.F85B2F9F |
| Database: |
BASE |
