On the intersection graph of random caps on a sphere

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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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