In this paper, we study the problem of asymptotic stability of continuous-time positive switched linear systems under both arbitrary and restricted switchings. It is well-known that asymptotic stability under arbitrary switching can be implied by several classes of strong common copositive Lyapunov functions (CLFs), i.e., functions whose derivative along the nontrivial system trajectories is negative. However, asymptotically stable positive switched systems may not admit strong common CLFs. The main contribution of this paper is to study the stability problem by requiring only weak common CLFs. Firstly, necessary and sufficient conditions are established for asymptotic stability under arbitrary switching, Among them, an easily verifiable graphical stability criterion, based on the connectivity of the digraphs associated with the subsystem matrices, is proposed. Secondly, we further relax the obtained graphical condition to derive a relaxed weak excitation condition for asymptotic stability under dwell-time switching. Finally, two examples are provided to illustrate the effectiveness of our theoretical results. (C) 2018 Elsevier Ltd. All rights reserved.