Graph classification aims to predict the label associated with a graph and is an important graph analytic task with widespread applications. Recently, graph neural networks (GNNs) have achieved state-of-the-art results on purely supervised graph classification by virtue of the powerful representation ability of neural networks. However, almost all of them ignore the fact that graph classification usually lacks reasonably sufficient labeled data in practical scenarios due to the inherent labeling difficulty caused by the high complexity of graph data. The existing semisupervised GNNs typically focus on the task of node classification and are incapable to deal with graph classification. To tackle the challenging but practically useful scenario, we propose a novel and general semisupervised GNN framework for graph classification, which takes full advantage of a slight amount of labeled graphs and abundant unlabeled graph data. In our framework, we train two GNNs as complementary views for collaboratively learning high-quality classifiers using both labeled and unlabeled graphs. To further exploit the view itself, we constantly select pseudo-labeled graph examples with high confidence from its own view for enlarging the labeled graph dataset and enhancing predictions on graphs. Furthermore, the proposed framework is investigated on two specific implementation regimes with a few labeled graphs and the extremely few labeled graphs, respectively. Extensive experimental results demonstrate the effectiveness of our proposed semisupervised GNN framework for graph classification on several benchmark datasets.