  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.
