Graph matching has important applications in pattern recognition and beyond. Current approaches predominantly adopt supervised learning, demanding extensive labeled data which can be limited or costly. Meanwhile, self-supervised learning methods for graph matching often require additional side information such as extra categorical information and input features, limiting their application to the general case. Moreover, designing the optimal graph augmentations for self-supervised graph matching presents another challenge to ensure robustness and efficacy. To address these issues, we introduce a novel Graph-centric Contrastive framework for Graph Matching (GCGM), capitalizing on a vast pool of graph augmentations for contrastive learning, yet without needing any side information. Given the variety of augmentation choices, we further introduce a Boosting-inspired Adaptive Augmentation Sampler (BiAS), which adaptively selects more challenging augmentations tailored for graph matching. Through various experiments, our GCGM surpasses state-of-the-art self-supervised methods across various datasets, marking a significant step toward more effective, efficient and general graph matching.