On a condition for the union of spherical caps to be connected

  • Authors:

  • H. Maehara

  • Affiliations:

  • College of Education, Ryukyu University, Okinawa 903-0213, Japan

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2003

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Abstract

Let c be a finite family of spherical caps of various sizes on a sphere in 3-space. A cap C ∈ c is called extremal if there is a great circle g such that the centers of those caps that intersect C lie in the same side of g, allowing some of them lie on g. We prove that if C contains no extremal cap, then the intersection graph of the caps in c is connected, and if furthermore every cap is smaller than a hemisphere, then the intersection graph is 2-connected. We also show that analogous assertions are no longer true in higher dimensions.