5-cycles in the complement of minimal prime graphs

Abstract: Minimal prime graphs (MPGs) are a special class of prime graphs (also known as Gruenberg-Kegel graphs) associated with finite solvable groups. A graph is an MPG if it has at least two vertices, is connected, its complement is triangle-free and 3-colorable, and the addition of an edge to the complement will violate triangle-freeness or 3-colorability. In this paper, we continue the study of the complements of MPGs focusing on their cycle structure. Our main result establishes that every edge in the complement of an MPG is contained in a 5-cycle. This finding is a much stronger form of an older result stating that every minimal prime graph complement contains at least one induced 5-cycle.