Minimal atlases of closed contact manifolds
  We study the minimal number C(M,\xi) of contact charts that one needs to cover a closed connected contact manifold (M,\xi). Our basic result is C(M,\xi) \le \dim M + 1.   We compute C(M,\xi) for all closed connected contact 3-manifolds:   C (M,\xi) = 2 if M = S^3 and \xi is tight,   3 if M = S^3 and \xi is overtwisted or if M = #_k (S^2 \times S^1),   4 otherwise.   We also show that on every sphere S^{2n+1} there exists a contact structure with C(S^{2n+1},\xi) \ge 3. 
Yuri Chekanov, Otto van Koert and Felix Schlenk