On Simmons’ “Campaign Graphs”

Abstract

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 $k\le 4$.

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.