Clique factors in pseudorandom graphs
An n-vertex graph is said to to be (p,ß)-bijumbled if for any vertex sets A,B¿V(G), we have e(A,B)=p|A||B|±ß |A||B| ¿ . We prove that for any r¿N =3 ¿ and c>0 there exists an e>0 such that any n-vertex (p,ß)-bijumbled graph with n¿rN, p>0, d(G)=cpn and ß=ep r-1 n contains a K r ¿ -factor. T...
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| Tipo de documento: | artigo |
| Data de publicação: | 2025 |
| País: | España |
| Recursos: | Universitat Politècnica de Catalunya (UPC) |
| Repositório: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglês |
| OAI Identifier: | oai:upcommons.upc.edu:2117/425707 |
| Acesso em linha: | https://hdl.handle.net/2117/425707 https://dx.doi.org/10.4171/JEMS/1388 |
| Access Level: | Acceso aberto |
| Palavra-chave: | Pseudorandom graphs Clique factors Extremal graph theory Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs |
| Resumo: | An n-vertex graph is said to to be (p,ß)-bijumbled if for any vertex sets A,B¿V(G), we have e(A,B)=p|A||B|±ß |A||B| ¿ . We prove that for any r¿N =3 ¿ and c>0 there exists an e>0 such that any n-vertex (p,ß)-bijumbled graph with n¿rN, p>0, d(G)=cpn and ß=ep r-1 n contains a K r ¿ -factor. This implies a corresponding result for the stronger pseudorandom notion of (n,d,¿)-graphs. For the case of triangle factors, that is, when r=3, this result resolves a conjecture of Krivelevich, Sudakov and Szabó from 2004 and it is tight due to a pseudorandom triangle-free construction of Alon. In fact, in this case even more is true: as a corollary to this result and a result of Han, Kohayakawa, Person and the author, we can conclude that the same condition of ß=o(p 2 n) actually guarantees that a (p,ß)-bijumbled graph G contains every graph on n vertices with maximum degree at most 2. |
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