Degree 2 vertices in minimal prime graph complements

TL;DR

This paper characterizes the structure of minimal prime graph complements with a degree 2 vertex, showing they form a specific class of graphs called reseminant, which are essentially 5-cycles with duplicated vertices.

Contribution

It proves that the presence of a degree 2 vertex in a minimal prime graph complement uniquely determines its structure as a reseminant graph.

Findings

Minimal prime graph complements with a degree 2 vertex are reseminant graphs.

Such graphs are essentially 5-cycles with duplicated vertices.

The structure is fully determined by the existence of a degree 2 vertex.

Abstract

Minimal prime graphs are connected graphs on at least two vertices whose complements satisfy the following conditions: triangle-freeness, 3-colorability, and edge-maximality with respect to the latter two properties. These graphs are prime graphs (or Gruenberg-Kegel graphs) of finite solvable groups with the maximum number of Frobenius actions among their Sylow subgroups, and as such minimal prime graph complements have been shown to be highly structured, including, for instance, the presence of induced 5-cycles. It is also known that the minimum degree of minimal prime graph complements is 2. In this note, we show that the existence of a degree 2 vertex in a minimal prime graph complement determines its whole structure: it is simply a 5-cycle with three vertices, exactly two of which are adjacent to each other, being duplicated finitely often. In particular, such graphs belong to a class of graphs known as reseminant.