TY - JOUR

T1 - On the Local Structure of Oriented Graphs – a Case Study in Flag Algebras

AU - Gilboa, Shoni

AU - Glebov, Roman

AU - Hefetz, Dan

AU - Linial, Nati

AU - Morgenstern, Avraham

N1 - Publisher Copyright:
© The authors.

PY - 2022

Y1 - 2022

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85135634990&partnerID=8YFLogxK

U2 - 10.37236/10694

DO - 10.37236/10694

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AN - SCOPUS:85135634990

SN - 1077-8926

VL - 29

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

IS - 3

M1 - 3

ER -