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Phase transition phenomena observed in most combinatorial search problems including constraint satisfaction problems (CSPs) are important for clarifying how structures make problem instances hard to solve. For the graph 3-colorability (3COL), which is one of the typical CSPs, the method to systematically generate hard problem instances by embedding original minimal unsolvable structures has been proposed, whereas many research reports are based on random generate-and-test approaches. In this paper, we extend the systematic method, enabling to generate a hard 3COL instances with a higher-order connectivity. We demonstrate that the computational cost to solve 3COL instances generated by our method is of an exponential order of the number of vertices by using a few coloring algorithms and constraint satisfaction algorithms.