Claim verification is a core module in automated fact-checking systems, tasked with determining claim veracity using retrieved evidence. This work presents KG-CRAFT, a novel knowledge graph-based contrastive reasoning method that enhances automatic claim verification by LLMs. Our approach first constructs a knowledge graph from claims and associated reports, then formulates contextually relevant contrastive questions based on the knowledge graph structure. These questions guide the distillation of evidence-based reports, which are synthesised into a concise summary for veracity assessment. Extensive evaluations on two real-world datasets (LIAR-RAW and RAWFC) demonstrate that our method achieves a new state-of-the-art in predictive performance. Comprehensive analyses validate in detail the effectiveness of our knowledge graph-based contrastive reasoning approach in improving LLMs’ fact-checking capabilities.