Cycles of a given length in tournaments

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21340%2F23%3A00361763" target="_blank" >RIV/68407700:21340/23:00361763 - isvavai.cz</a>
Alternative codes found
RIV/00216224:14330/23:00130429
Result on the web
<a href="http://hdl.handle.net/10467/107776" target="_blank" >http://hdl.handle.net/10467/107776</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1016/j.jctb.2022.07.007" target="_blank" >10.1016/j.jctb.2022.07.007</a>

Result language
angličtina
Original language name
Cycles of a given length in tournaments
Original language description
We study the asymptotic behavior of the maximum number of directed cycles of a given length in a tournament: let c(l) be the limit of the ratio of the maximum number of cycles of length l in an n-vertex tournament and the expected number of cycles of length l in the random n-vertex tournament, when n tends to infinity. It is well-known that c(3)=1 and c(4)=4/3. We show that c(l)=1 if and only if l is not divisible by four, which settles a conjecture of Bartley and Day. If l is divisible by four, we show that 1+2(2/π)^l <= c(l) <= 1+(2/π+o(1))^l and determine the value c(l) exactly for l=8. We also give a full description of the asymptotic structure of tournaments with the maximum number of cycles of length l when l is not divisible by four or l=4 or l=8.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2023
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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