TY - CONF

T1 - A generalized Turán problem and its applications

AU - Gishboliner, Lior

AU - Shapira, Asaf

N1 - Publisher Copyright:
© 2019 16th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, CTW 2018 - Proceedings of the Workshop.

PY - 2019

Y1 - 2019

N2 - The investigation of conditions guaranteeing the appearance of cycles of certain lengths is one of the most well-studied topics in graph theory. In this paper we consider a problem of this type which asks, for fixed integers ` and k, how many copies of the k-cycle guarantee the appearance of an `-cycle? Extending previous results of Bollobás-Gyori-Li and Alon-Shikhelman, we fully resolve this problem by giving tight (or nearly tight) bounds for all values of ` and k. We also present a somewhat surprising application of the above mentioned estimates to the study of graph property testing. Prior to this work, all bounds for the query complexity of testing graph properties were either polynomial or there was a tower-type gap between the best known upper and lower bounds. We fill this gap by showing that for every super-polynomial function f(e), there is a monotone graph property P, such that the query complexity of the optimal one-sided-error e-tester for P is precisely given by f(e). We thus obtain the first examples of tight super-polynomial bounds for the one-sided-error query complexity of graph properties. A special case of this result resolves a problem of Alon and the second author, while another special case partially resolves a problem of Goldreich.

AB - The investigation of conditions guaranteeing the appearance of cycles of certain lengths is one of the most well-studied topics in graph theory. In this paper we consider a problem of this type which asks, for fixed integers ` and k, how many copies of the k-cycle guarantee the appearance of an `-cycle? Extending previous results of Bollobás-Gyori-Li and Alon-Shikhelman, we fully resolve this problem by giving tight (or nearly tight) bounds for all values of ` and k. We also present a somewhat surprising application of the above mentioned estimates to the study of graph property testing. Prior to this work, all bounds for the query complexity of testing graph properties were either polynomial or there was a tower-type gap between the best known upper and lower bounds. We fill this gap by showing that for every super-polynomial function f(e), there is a monotone graph property P, such that the query complexity of the optimal one-sided-error e-tester for P is precisely given by f(e). We thus obtain the first examples of tight super-polynomial bounds for the one-sided-error query complexity of graph properties. A special case of this result resolves a problem of Alon and the second author, while another special case partially resolves a problem of Goldreich.

UR - http://www.scopus.com/inward/record.url?scp=85075912874&partnerID=8YFLogxK

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AN - SCOPUS:85075912874

SP - 128

EP - 131

T2 - 16th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, CTW 2018

Y2 - 18 June 2018 through 20 June 2018

ER -