Hereditary properties of combinatorial structures: posets and oriented graphs

József Balogh, Béla Bollobás, Robert Morris

Research output: Contribution to journalArticlepeer-review

Abstract

A hereditary property of combinatorial structures is a collection of structures (e.g., graphs, posets) which is closed under isomorphism, closed under taking induced substructures (e.g., induced subgraphs), and contains arbitrarily large structures. Given a property □P1 we write Pn for the collection of distinct (i.e., non-isomorphic) structures in a property V with n vertices, and call the function n > □ |Pn| the speed (or unlabeled speed) of P. Also, we write Pn for the collection of distinct labeled structures in V with vertices labeled 1, . . . , n, and call the function ni □ |Pn| the labeled speed of V. The possible labeled speeds of a hereditary property of graphs have been extensively studied, and the aim of this article is to investigate the possible speeds of other combinatorial structures, namely posets and oriented graphs. More precisely, we show that (for sufficiently large n), the labeled speed of a hereditary property of posets is either 1, or exactly a polynomial, or at least 2n - 1. We also show that there is an initial jump in the possible unlabeled speeds of hereditary properties of posets, tournaments, and directed graphs, from bounded to linear speed, and give a sharp lower bound on the possible linear speeds in each case.

Original languageEnglish (US)
Pages (from-to)311-332
Number of pages22
JournalJournal of Graph Theory
Volume56
Issue number4
DOIs
StatePublished - Dec 2007

Keywords

  • Hereditary property
  • Poset
  • Tournament

ASJC Scopus subject areas

  • Geometry and Topology

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