Knowledge graph construction is aimed at storing and representing the knowledge of the objective world in a structured form. Existing methods for automatic construction of knowledge graphs have problems such as difficulty in understanding potential semantics and low precision. The emergence of Large Language Models (LLMs) provides an effective way for automatic knowledge graph construction. However, using LLMs as automatic knowledge graph construction engines relies on the embedding of schema layers, which brings challenges to the input length of LLMs. In this paper, we present a framework for Adaptive Construction of Knowledge Graph by leveraging the exceptional generation capabilities of LLMs and the latent relational semantic information of triples, named ACKG-LLM. Our proposed framework divides the knowledge graph construction task into three subtasks within a unified pipeline: triple extraction of open information, additional relational semantic information embedding and knowledge graph normalization based on schema-level embedding. The framework can construct knowledge graphs in different domains, making up for the defects of existing frameworks that need to retrain and fine-tune the internal model. Extensive experiments demonstrate that our proposed ACKG-LLM performs favorably against representative methods on the REBEL and WiKi-NRE datasets. The code is available athttps://github.com/KustTeamWQW/ACKG-LLM