TY - JOUR

T1 - Extremal results for odd cycles in sparse pseudorandom graphs

AU - Aigner-Horev, Elad

AU - Hàn, Hiệp

AU - Schacht, Mathias

N1 - Publisher Copyright:
© 2014, János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg.

PY - 2014/7/3

Y1 - 2014/7/3

N2 - We consider extremal problems for subgraphs of pseudorandom graphs. For graphs F and Г the generalized Turán density πF(Г) denotes the relative density of a maximum subgraph of Г, which contains no copy of F. Extending classical Turán type results for odd cycles, we show that πF(Г)=1/2 provided F is an odd cycle and Г is a sufficiently pseudorandom graph.In particular, for (n,d,λ)-graphs Г, i.e., n-vertex, d-regular graphs with all non-trivial eigenvalues in the interval [−λ,λ], our result holds for odd cycles of length ℓ, provided (Formula presented.) Up to the polylog-factor this verifies a conjecture of Krivelevich, Lee, and Sudakov. For triangles the condition is best possible and was proven previously by Sudakov, Szabó, and Vu, who addressed the case when F is a complete graph. A construction of Alon and Kahale (based on an earlier construction of Alon for triangle-free (n,d;λ)-graphs) shows that our assumption on Г is best possible up to the polylog-factor for every odd ℓ≥5.

AB - We consider extremal problems for subgraphs of pseudorandom graphs. For graphs F and Г the generalized Turán density πF(Г) denotes the relative density of a maximum subgraph of Г, which contains no copy of F. Extending classical Turán type results for odd cycles, we show that πF(Г)=1/2 provided F is an odd cycle and Г is a sufficiently pseudorandom graph.In particular, for (n,d,λ)-graphs Г, i.e., n-vertex, d-regular graphs with all non-trivial eigenvalues in the interval [−λ,λ], our result holds for odd cycles of length ℓ, provided (Formula presented.) Up to the polylog-factor this verifies a conjecture of Krivelevich, Lee, and Sudakov. For triangles the condition is best possible and was proven previously by Sudakov, Szabó, and Vu, who addressed the case when F is a complete graph. A construction of Alon and Kahale (based on an earlier construction of Alon for triangle-free (n,d;λ)-graphs) shows that our assumption on Г is best possible up to the polylog-factor for every odd ℓ≥5.

KW - 05C35

KW - 05C80

KW - 05D40

UR - http://www.scopus.com/inward/record.url?scp=84894433979&partnerID=8YFLogxK

U2 - 10.1007/s00493-014-2912-y

DO - 10.1007/s00493-014-2912-y

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AN - SCOPUS:84894433979

SN - 0209-9683

VL - 34

SP - 379

EP - 406

JO - Combinatorica

JF - Combinatorica

IS - 4

ER -