Intersection Graphs of Spherical Caps

or a family \({\mathcal C}=\{C_1,C_2,\dots , C_N\}\) of spherical caps on \({\mathbb S}\) , its intersection graph \(\Omega ({\mathcal C})\) is defined as the graph whose vertices are the centers \(x_i\) of the caps \(C_i\) , \(i=1,2,\dots , N\) , and two vertices \(x_i, x_j\) are adjacent if and only if \(C_i\cap C_j\ne \emptyset \) . We present a sufficient condition for the connectedness of \(\Omega ({\mathcal C})\) .