Fault detection and localization are vital for ensuring the stability of data center networks (DCNs). Specifically, adaptive fault diagnosis is deemed a fundamental technology in achieving the fault tolerance of systems. The highly scalable data center network (HSDC) is a promising structure of server-centric DCNs, as it exhibits the capacity for incremental scalability, coupled with the assurance of low cost and energy consumption, low diameter, and high bisection width. In this paper, we first determine that both the connectivity and diagnosability of the m-dimensional complete HSDC, denoted by HSDCm(m), are m. Further, we propose an efficient adaptive fault diagnosis algorithm to diagnose an HSDCm(m) within three test rounds, and at most N+4m(m−2) tests with m≥3 (resp. at most nine tests with m=2), where N=m·2m is the total number of nodes in HSDCm(m). Our experimental outcomes demonstrate that this diagnosis scheme of HSDC can achieve complete diagnosis and significantly reduce the number of required tests.