# On the intersection graph of random caps on a sphere

## Abstract

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\$.

## Journal

• European Journal of Combinatorics

European Journal of Combinatorics 25(5), 707-718, 2004-07

Elsevier Science B.V., Amsterdam

## Codes

• NII Article ID (NAID)
120001371781
• NII NACSIS-CAT ID (NCID)
AA00181294
• Text Lang
ENG
• Article Type
journal article
• ISSN
0195-6698
• Data Source
IR

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