Degree Conditions Forcing Directed Cycles

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21340%2F23%3A00365664" target="_blank" >RIV/68407700:21340/23:00365664 - isvavai.cz</a>
Result on the web
<a href="http://hdl.handle.net/10467/108535" target="_blank" >http://hdl.handle.net/10467/108535</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1093/imrn/rnac114" target="_blank" >10.1093/imrn/rnac114</a>

Result language
angličtina
Original language name
Degree Conditions Forcing Directed Cycles
Original language description
Caccetta-Haggkvist conjecture is a longstanding open problem on degree conditions that force an oriented graph to contain a directed cycle of a bounded length. Motivated by this conjecture, Kelly, Kuhn, and Osthus initiated a study of degree conditions forcing the containment of a directed cycle of a given length. In particular, they found the optimal minimum semidegree, that is, the smaller of the minimum indegree and the minimum outdegree, which forces a large oriented graph to contain a directed cycle of a given length not divisible by 3, and conjectured the optimal minimum semidegree for all the other cycles except the directed triangle. In this paper, we establish the best possible minimum semidegree that forces a large oriented graph to contain a directed cycle of a given length divisible by 3 yet not equal to 3, hence fully resolve the conjecture by Kelly, Kuhn, and Osthus. We also find an asymptotically optimal semidegree threshold of any cycle with a given orientation of its edges with the sole exception of a directed triangle.
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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