In sprite the state-of-the-art, significantly reducing carbon footprint (CF) in communications systems remains urgent. We address this challenge in the context of edge computing. The carbon intensity of electricity supply largely varies spatially as well as temporally. This, together with energy sharing via a battery management system (BMS), justifies the potential of CF-oriented task offloading, by redistributing the computational tasks in time and space. In this paper, we consider optimal task scheduling and offloading, as well as battery charging to minimize the total CF. We formulate this CF minimization problem as an integer linear programming model. However, we demonstrate that, via a graph-based reformulation, the problem can be cast as a minimum-cost flow problem. This finding reveals that global optimum can be admitted in polynomial time. Numerical results using real-world data show that optimization can reduce up to 83.3% of the total CF.