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
M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???
AN - SCOPUS:84894433979
SN - 0209-9683
VL - 34
SP - 379
EP - 406
JO - Combinatorica
JF - Combinatorica
IS - 4
ER -