Graph theory is an interdisciplinary field of study that has various applications in mathematical modeling and computer science. Research in graph theory depends on the creation of not only theorems but also conjectures. Conjecture-refuting algorithms attempt to refute conjectures by searching for counterexamples to those conjectures, often by maximizing certain score functions on graphs. This study proposes a novel conjecture-refuting algorithm, referred to as the adaptive Monte Carlo search (AMCS) algorithm, obtained by modifying the Monte Carlo tree search algorithm. Evaluated based on its success in finding counterexamples to several graph theory conjectures, AMCS outperforms existing conjecture-refuting algorithms. The algorithm is further utilized to refute six open conjectures, two of which were chemical graph theory conjectures formulated by Liu et al. in 2021 and four of which were formulated by the AutoGraphiX computer system in 2006. Finally, four of the open conjectures are strongly refuted by generalizing the counterexamples obtained by AMCS to produce a family of counterexamples. It is expected that the algorithm can help researchers test graph-theoretic conjectures more effectively.

翻译：图论是一门跨学科研究领域，在数学建模和计算机科学中具有广泛应用。图论研究不仅依赖定理的建立，更依赖于猜想的提出。猜想反例算法通过寻找反例来证伪猜想，通常通过最大化图上某些评分函数实现。本研究提出一种新型猜想反例算法——自适应蒙特卡洛搜索（AMCS）算法，该算法通过对蒙特卡洛树搜索算法进行改进而获得。基于其在多个图论猜想反例搜索中的表现进行评估，AMCS算法优于现有猜想反例算法。该算法进一步被用于证伪六个未解猜想，其中两个是Liu等人于2021年提出的化学图论猜想，另外四个由AutoGraphiX计算机系统于2006年提出。最后，通过将AMCS算法获得的特定反例推广为反例族，四个未解猜想得到强证伪。该算法有望帮助研究人员更有效地检验图论猜想。