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A multi-cell cluster-free NOMA framework is proposed, where both intra-cell and inter-cell interference are jointly mitigated via flexible cluster-free successive interference cancellation (SIC) and coordinated beamforming design. The joint design problem is formulated to maximize the system sum rate while satisfying the SIC decoding requirements and users' minimum data rate requirements. To address this highly complex and coupling non-convex mixed integer nonlinear programming (MINLP), a novel distributed auto-learning graph neural network (AutoGNN) architecture is proposed to alleviate the overwhelming information exchange burdens among base stations (BSs). The proposed AutoGNN can train the GNN model weights whilst automatically optimizing the GNN architecture, namely the GNN network depth and message embedding sizes, to achieve communication-efficient distributed scheduling. Based on the proposed architecture, a bi-level AutoGNN learning algorithm is further developed to efficiently approximate the hypergradient in model training. It is theoretically proved that the proposed bi-level AutoGNN learning algorithm can converge to a stationary point. Numerical results reveal that: 1) the proposed cluster-free NOMA framework outperforms the conventional cluster-based NOMA framework in the multi-cell scenario; and 2) the proposed AutoGNN architecture significantly reduces the computation and communication overheads compared to the conventional convex optimization-based methods and the conventional GNNs with fixed architectures.