Personalizing language models by effectively incorporating user interaction history remains a central challenge in the development of adaptive AI systems. While large language models (LLMs) combined with Retrieval-Augmented Generation (RAG) have improved factual accuracy, they often lack structured memory and fail to scale in complex, long-term interactions. To address this, we propose a flexible external memory framework based on knowledge graphs, automatically constructed and updated by the LLM itself, and capable of encoding information in multiple formats-including nodes, triplets, higher-order propositions, and episodic traces. Building upon the AriGraph architecture, we introduce a novel hybrid graph design that supports both standard edges and two types of hyperedges, enabling rich and dynamic semantic and temporal representations. Our framework also supports diverse retrieval mechanisms, including A*, water-circle propagation, beam search, and hybrid methods, making it adaptable to different datasets and LLM capacities. We evaluate our system on three benchmarks-TriviaQA, HotpotQA, and DiaASQ-demonstrating that different memory and retrieval configurations yield optimal performance depending on the task. Additionally, we extend the DiaASQ benchmark with temporal annotations and internally contradictory statements, showing that our system remains robust and effective in managing temporal dependencies and context-aware reasoning.