We prove that there is a unique graph with four edges which is the Gruenberg-Kegel graph of a solvable cut group. This contributes to the classification of the Gruenberg-Kegel graphs of solvable cut groups initiated in [A. B\"achle, A. Kiefer, S. Maheshwary, and A. del R\'io. "Gruenberg-Kegel graphs: Cut groups, rational groups and the prime graph question" Forum Mathematicum, vol. 35, no. 2, 2023, pp. 409-429. https://doi.org/10.1515/forum-2022-0086].
@article{arxiv.2604.01817,
title = {Gruenberg-Kegel graphs with four edges of finite solvable cut groups},
author = {Sara C. Debón and Diego García-Lucas and Ángel del Río},
journal= {arXiv preprint arXiv:2604.01817},
year = {2026}
}