Carl Hierholzer (2 October 1840 – 13 September 1871) was a

German German(s) may refer to: * Germany (of or related to) **Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ger ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

Biography

Hierholzer studied mathematics in

Karlsruhe Karlsruhe ( , , ; South Franconian: ''Kallsruh'') is the third-largest city of the German state (''Land'') of Baden-Württemberg after its capital of Stuttgart and Mannheim, and the 22nd-largest city in the nation, with 308,436 inhabitants. ...

, and he got his

Ph.D. A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin: or ') is the most common degree at the highest academic level awarded following a course of study. PhDs are awarded for programs across the whole breadth of academic fields. Because it is a ...

from

Ruprecht-Karls-Universität Heidelberg } Heidelberg University, officially the Ruprecht Karl University of Heidelberg, (german: Ruprecht-Karls-Universität Heidelberg; la, Universitas Ruperto Carola Heidelbergensis) is a public research university in Heidelberg, Baden-Württemberg, ...

in 1865. His

advisor was

Ludwig Otto Hesse Ludwig Otto Hesse (22 April 1811 – 4 August 1874) was a German mathematician. Hesse was born in Königsberg, Prussia, and died in Munich, Bavaria. He worked mainly on algebraic invariants, and geometry. The Hessian matrix, the Hesse norm ...

(1811–1874). In 1870 Hierholzer wrote his

habilitation Habilitation is the highest university degree, or the procedure by which it is achieved, in many European countries. The candidate fulfills a university's set criteria of excellence in research, teaching and further education, usually including a ...

about conic sections (title: ''Ueber Kegelschnitte im Raum'') in Karlsruhe, where he later became a

Privatdozent ''Privatdozent'' (for men) or ''Privatdozentin'' (for women), abbreviated PD, P.D. or Priv.-Doz., is an academic title conferred at some European universities, especially in German-speaking countries, to someone who holds certain formal qualific ...

. Hierholzer proved that a connected

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...

has an

Eulerian trail In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices). Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends ...

if and only if exactly zero or two of its vertices have an odd degree. This result had been given, with no proof of the 'if' part, by

Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

in 1736. Hierholzer apparently presented his work to a circle of fellow mathematicians not long before his premature death in 1871. A colleague then arranged for its posthumous publication in a paper that appeared in 1873.

References

* C. Hierholzer: ''Ueber Kegelschnitte im Raume''. (Habilitation in Karlsruhe.) Mathematische Annalen II (1870), 564–586

* C. Hierholzer: ''Ueber eine Fläche der vierten Ordnung''. Mathematische Annalen IV (1871), 172–180

* C. Hierholzer: ''Über die Möglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren''. Mathematische Annalen VI (1873), 30–32

* Barnett, Janet Hein
Early Writings on Graph Theory: Euler Circuits and The Königsberg Bridge Problem
{{DEFAULTSORT:Hierholzer, Carl 1840 births 1871 deaths Scientists from Freiburg im Breisgau People from the Grand Duchy of Baden 19th-century German mathematicians