5-cycles in the complement of minimal prime graphs

TL;DR

This paper investigates the cycle structure of the complements of minimal prime graphs, proving that every edge in such complements is contained in a 5-cycle, strengthening previous results about their cycle properties.

Contribution

It establishes that all edges in the complement of a minimal prime graph are part of a 5-cycle, a significant strengthening of earlier findings.

Findings

Every edge in the complement of an MPG is in a 5-cycle.

The complement of an MPG contains no edges outside these 5-cycles.

This enhances understanding of the cycle structure in MPG complements.

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.