Degree 2 vertices in minimal prime graph complements

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.