Federated AUC maximization is a powerful approach for learning from imbalanced data in federated learning (FL). However, existing methods typically assume full client availability, which is rarely practical. In real-world FL systems, clients often participate in a cyclic manner: joining training according to a fixed, repeating schedule. This setting poses unique optimization challenges for the non-decomposable AUC objective. This paper addresses these challenges by developing and analyzing communication-efficient algorithms for federated AUC maximization under cyclic client participation. We investigate two key settings: First, we study AUC maximization with a squared surrogate loss, which reformulates the problem as a nonconvex-strongly-concave minimax optimization. By leveraging the Polyak-Łojasiewicz (PL) condition, we establish a state-of-the-art communication complexity of $\widetilde{O}(1/ε^{1/2})$ and iteration complexity of $\widetilde{O}(1/ε)$. Second, we consider general pairwise AUC losses. We establish a communication complexity of $O(1/ε^3)$ and an iteration complexity of $O(1/ε^4)$. Further, under the PL condition, these bounds improve to communication complexity of $\widetilde{O}(1/ε^{1/2})$ and iteration complexity of $\widetilde{O}(1/ε)$. Extensive experiments on benchmark tasks in image classification, medical imaging, and fraud detection demonstrate the superior efficiency and effectiveness of our proposed methods.