Regular sets in circulant graphs with degree 3

Let Gamma = (V, E) be a graph and a, b nonnegative integers. An (a, b)-regular set in Gamma is a nonempty proper subset D of V such that every vertex in D has exactly a neighbours in D and every vertex in V \ D has exactly b neighbours in D. In this paper, we obtain a necessary and sufficient condition for the existence of (a, b)-regular sets in connected circulant graphs with degree 3, for all possible values a, b. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.