Networks and Spanning Trees: The Juxtaposition of Prüfer and Borůvka

Abstract

This paper outlines a method for teaching topics in undergraduate mathematics or computer science via historical curricular modules. The contents of one module, “Networks and Spanning Trees,” are discussed from the original work of Arthur Cayley, Heinz Prüfer, and Otakar Borůvka that motivates the enumeration and application of trees in graph theory. Cayley correctly identifies a pattern for the number of (labeled) trees on n fixed vertices. Prüfer’s paper provides a rigorous verification of this pattern, whereas Borůvka’s paper offers one of the first algorithms for finding a minimal spanning tree over the domain of labeled trees. These latter two papers in juxtaposition offer a pleasing confluence of concepts and applications, written verbally before the modern terminology of graph theory had been formulated.