TY - JOUR

T1 - On the Edge Distribution in Triangle-free Graphs

AU - Krivelevich, Michael

PY - 1995/3

Y1 - 1995/3

N2 - Two problems on the edge distribution in triangle-free graphs are considered: (1) Given an 0 < α < 1. Find the largest β = β(α) such that for infinitely many n there exists a triangle-free graph G on n vertices in which every αn vertices span at least βn2 edges. This problem was considered by Erdös, Faudree, Rousseau, and Schelp in (J. Combin. Theory Ser. B45 (1988), 111-120). Here we extend and improve their results, proving in particular the bound β < 1/36 for α = 1/2; (2) How much does the edge distribution in a triangle-free graph G on n vertices deviate from the uniform edge distribution in a typical (random) graph on n vertices with the same number of edges? We give quantitative expressions for this deviation.

AB - Two problems on the edge distribution in triangle-free graphs are considered: (1) Given an 0 < α < 1. Find the largest β = β(α) such that for infinitely many n there exists a triangle-free graph G on n vertices in which every αn vertices span at least βn2 edges. This problem was considered by Erdös, Faudree, Rousseau, and Schelp in (J. Combin. Theory Ser. B45 (1988), 111-120). Here we extend and improve their results, proving in particular the bound β < 1/36 for α = 1/2; (2) How much does the edge distribution in a triangle-free graph G on n vertices deviate from the uniform edge distribution in a typical (random) graph on n vertices with the same number of edges? We give quantitative expressions for this deviation.

UR - http://www.scopus.com/inward/record.url?scp=0037600748&partnerID=8YFLogxK

U2 - 10.1006/jctb.1995.1018

DO - 10.1006/jctb.1995.1018

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AN - SCOPUS:0037600748

SN - 0095-8956

VL - 63

SP - 245

EP - 260

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

IS - 2

ER -