2024-03-29T15:23:46Z
https://u-ryukyu.repo.nii.ac.jp/oai
oai:u-ryukyu.repo.nii.ac.jp:02000876
2023-08-03T05:27:11Z
1642838163960:1642838338003
1642838403551:1642838405037
On the intersection graph of random caps on a sphere
Maehara, Hiroshi
Drop $N$ spherical caps, each of area $4\pi ・ $p(N)$, at random on the surface of a unit sphere, and let $G\sb{p}$ denote the intersection graphs of these random caps. Among others, we prove the following: (1) If $N(N\sb{p})\sp{n-1}\to\0$ as $N \to\infty$, then ${\rm Pr}(G\sb{p}\text{ has no component of order }\geq n)\to1$, while if $N(N\sb{p})\sp(n-1) \to\ infty$ then ${\rm Pr}(G\sb{p}\text{ has an $n$-clique})\to1$ as $N\to\infty$. (2) If, $p<(1-\varepsilon)\log N/4N$, $\varepsilon>0$ then ${\rm Pr}(\delta=0)\to1$, while if $p>(1+\varepsilon)\log N/4N$ then for any positive integer $n$, ${\rm Pr}(\delta\geq n)\to1$ as $ N\to\infty$, where $\delta$ denotes the minimum degree of $G\sb{p}$. (3) If $p=(\log N+x)/4N$ then the number of isolated vertices of $G\sb{p}$ is asymptotically $(N\to\infty)$ distributed according to Poisson distribution with mean $e\sp{-x}$. (4) If $p>(1+\varepsilon)\log N/2N$, then ${\rm Pr}(G\sb{p}\text{ is $2$-connected})\to 1$ as $N\to\infty$.
論文
http://purl.org/coar/resource_type/c_6501
Elsevier Science B.V., Amsterdam
2004-07
AM
http://hdl.handle.net/20.500.12000/149
01956698
AA00181294
European Journal of Combinatorics
5
25
718
707
eng
http://www.sciencedirect.com/science/journal/01956698
10.1016/j.ejc.2003.10.005
open access