Large Language Models have recently demonstrated remarkable potential in graph-based tasks by integrating natural language understanding with structural reasoning. However, when applied to dynamic graphs—networks that evolve over time, LLMs exhibit a surprising and counterintuitive phenomenon: their accuracy in answering the fundamental question when two nodes are linked is only around 50%. Since recognizing edge existence is one of the most basic graph understanding tasks, this fundamental shortcoming undermines critical graph operations such as triadic closure detection and degree calculation. We define this phenomenon as the hallucination problem in dynamic graphs. In response to these challenges, we introduce two novel mechanisms. First, the Existence Fine-tuning Mechanism explicitly trains LLMs to recognize and retain the persistent presence of edges in dynamic graphs. Second, the Edge Reduction Mechanism enforces a strategy that decomposes complex reasoning tasks into simpler, more reliable edge existence queries. To further enhance reasoning capabilities in smaller LLMs, we propose a Teacher-Forcing Distillation Strategy that leverages high-quality decomposition strategies generated by a larger model. Extensive experiments on multiple models first demonstrate hallucination is common in models around 7B. Further experiments also demonstrate that our methods substantially mitigate hallucinations and achieve state-of-the-art performance. In summary, our findings reveal a critical limitation in current LLMs’ handling of dynamic graphs and provide a robust framework and effective solutions for advancing dynamic graph reasoning capabilities in these models. All the codes can be accessed via this link ( https://anonymous.4open.science/r/DyG-Hallucination-6CCB ).