5-cycles in the complement of minimal prime graphs
Micah Dorton, Thomas Michael Keller, Ryan Tang, Justin Yu
TL;DR
This work addresses the cycle structure of the complement of minimal prime graphs (MPGCs) in finite solvable groups, proving a strengthening of previous results by showing that every edge of an MPGC lies in a $5$-cycle. It develops a constructive framework around the parametric subgraph family $\Gamma(m,n,k,l,x,y)$ and analyzes valid colorings of $7$-cycles to drive an inductive argument that either yields a $5$-cycle for every edge or forces an impossible $\Gamma$-type expansion. The main theorem situates MPGCs within triangle-free graphs in which each edge participates in a $5$-cycle and links this structure to Frobenius digraphs and Sylow configurations, with concrete illustrations from the Petersen family. The approach highlights potential pathways toward a broader classification of MPGCs and aligns with known examples, suggesting deep connections between group-theoretic prime graphs and classic triangle-free graph families.
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.