On the Local Structure of Oriented Graphs – a Case Study in Flag Algebras

Shoni Gilboa, Roman Glebov, Dan Hefetz*, Nati Linial, Avraham Morgenstern

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

Let G be an n-vertex oriented graph. Let t(G) (respectively i(G)) be the probability that a random set of 3 vertices of G spans a transitive triangle (respectively an independent set). We prove that t(G) + i(G) ≥19− on(1). Our proof uses the method of flag algebras that we supplement with several steps that make it more easily comprehensible. We also prove a stability result and an exact result. Namely, we describe an extremal construction, prove that it is essentially unique, and prove that if H is sufficiently far from that construction, then t(H) + i(H) is significantly larger than19. We go to greater technical detail than is usually done in papers that rely on flag algebras. Our hope is that as a result this text can serve others as a useful introduction to this powerful and beautiful method.

Original languageAmerican English
JournalElectronic Journal of Combinatorics
Volume29
Issue number3
DOIs
StatePublished - 2022

Bibliographical note

Funding Information:
Supported by ISF grant 822/18.

Publisher Copyright:
© The authors.

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