On Simmons’ “Campaign Graphs”

In a recent paper, Gustavus J. Simmons introduced a new class of combinatorial-geometric objects he called “campaign graphs”. A \(k\)-campaign graph is a collection of points and segments such that each segment contains precisely \(k\) of the points, and each point is the endpoint of precisely one segment. Among other results, Simmons proved the existence of infinitely many critical \(k\)-campaign graphs for \(k \leq 4\).

The main aim of this note is to show that Simmons’ result holds for \(k = 5\) and \(6\) as well, thereby providing proofs, amplifications and a correction for statements of this author which Dr. Simmons was kind enough to include in a postscript to his paper.