On the Minimum Order of Extremal Graphs to have a Prescribed Girth
We show that any n‐vertex extremal graph G without cycles of length at most k has girth exactly $k+1$ if $k\ge 6$ and $n>(2(k-2)^{k-2}+k-5)/(k-3)$. This result provides an improvement of the asymptotical known result by Lazebnik and Wang [J. Graph Theory, 26 (1997), pp. 147–153] who proved that t...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2007 |
| País: | España |
| Institución: | Universidad de Sevilla (US) |
| Repositorio: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:idus.us.es:11441/57152 |
| Acceso en línea: | http://hdl.handle.net/11441/57152 https://doi.org/10.1137/060656747 |
| Access Level: | acceso abierto |
| Palabra clave: | extremal graphs girth forbidden cycles cages |
| Sumario: | We show that any n‐vertex extremal graph G without cycles of length at most k has girth exactly $k+1$ if $k\ge 6$ and $n>(2(k-2)^{k-2}+k-5)/(k-3)$. This result provides an improvement of the asymptotical known result by Lazebnik and Wang [J. Graph Theory, 26 (1997), pp. 147–153] who proved that the girth is exactly $k+1$ if $k\ge 12$ and $n\ge 2^{a^2+a+1}k^a$, where $a=k-3-\lfloor(k-2)/4\rfloor$. Moreover, we prove that the girth of G is at most $k+2$ if $n>(2(t-2)^{k-2}+t-5)/(t-3)$, where $t=\lceil (k+1)/2\rceil\ge 4$. In general, for $k\ge 5$ we show that the girth of G is at most $2k-4$ if $n\ge 2k-2$. |
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