Heiko Harborth
Encyclopedia
Heiko Harborth is Professor of Mathematics
at Braunschweig University of Technology, 1975–present, and author of more than 188 mathematical publications. His work is mostly in the areas of number theory
, combinatorics
and discrete geometry
, including graph theory
.
Harborth is a member of the New York Academy of Sciences
, Braunschweigische Wissenschaftliche Gesellschaft, the Institute of Combinatorics and its Applications
, and many other mathematical societies.
Harborth currently sits on the editorial boards of Fibonacci Quarterly
, Geombinatorics
, Integers: Electronic Journal of Combinatorial Number Theory. He served as an editor of Mathematische Semesterberichte from 1988 to 2001.
Harborth was a joint recipient (with Stephen Milne) of the 2007 Euler Medal
.
Harborth's research ranges across the subject areas of combinatorics
, graph theory
, discrete geometry
, and number theory
.
In 1974, Harborth solved the unit coin graph problem, determining the maximum number of edges possible in a unit coin graph on n vertices.
In 1986, Harborth presented the graph that would bear his name, the Harborth Graph. It is the smallest known example of a 4-regular
matchstick graph
. It has 104 edges and 52 vertices.
In connection with the Happy Ending problem
, Harborth showed that, for every finite set of ten or more points in general position in the plane, some five of them form a convex pentagon that does not contain any of the other points.
Harborth's Conjecture posits that every planar graph admits a straight-line embedding in the plane where every edge has integer length. This open question is a stronger version of Fáry's theorem
. It is known to be true for cubic graph
s.
In number theory
, the Stolarsky–Harborth constant is named for Harborth, along with Kenneth Stolarsky.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
at Braunschweig University of Technology, 1975–present, and author of more than 188 mathematical publications. His work is mostly in the areas of number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
and discrete geometry
Discrete geometry
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles,...
, including graph theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
.
Career
Harborth has been an instructor or professor at Braunschweig University of Technology since studying there and receiving his PhD in 1965 under Hans-Joachim Kanold.Harborth is a member of the New York Academy of Sciences
New York Academy of Sciences
The New York Academy of Sciences is the third oldest scientific society in the United States. An independent, non-profit organization with more than members in 140 countries, the Academy’s mission is to advance understanding of science and technology...
, Braunschweigische Wissenschaftliche Gesellschaft, the Institute of Combinatorics and its Applications
Institute of Combinatorics and its Applications
The Institute of Combinatorics and its Applications is an international scientific organization. It was formed in 1990 and is based in Winnipeg, Canada.-Aim and membership:...
, and many other mathematical societies.
Harborth currently sits on the editorial boards of Fibonacci Quarterly
Fibonacci Quarterly
As the primary publication of The Fibonacci Association, The Fibonacci Quarterly provides a focus for worldwide interest in the Fibonacci number sequence and related mathematics. Published since 1963, its founding editors were Verner Emil Hoggatt, Jr. and Alfred Brousseau...
, Geombinatorics
Geombinatorics
Geombinatorics is a mathematical research journal founded by Alexander Soifer and published by the University of Colorado, USA, since 1991 under an international board of editors...
, Integers: Electronic Journal of Combinatorial Number Theory. He served as an editor of Mathematische Semesterberichte from 1988 to 2001.
Harborth was a joint recipient (with Stephen Milne) of the 2007 Euler Medal
Euler Medal
The Euler Medal, named after the 18th century mathematician Leonhard Euler, is an honor awarded annually by the Institute of Combinatorics and its Applications to a member with a distinguished lifetime contribution to combinatorial research who are still active in research.-Laureates:* 2008: Gabor...
.
Mathematical work
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
, graph theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
, discrete geometry
Discrete geometry
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles,...
, and number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
.
In 1974, Harborth solved the unit coin graph problem, determining the maximum number of edges possible in a unit coin graph on n vertices.
In 1986, Harborth presented the graph that would bear his name, the Harborth Graph. It is the smallest known example of a 4-regular
Regular graph
In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree of each vertex are equal to each other...
matchstick graph
Matchstick graph
In geometric graph theory, a branch of mathematics, a matchstick graph is a graph that can be drawn in the plane in such a way that its edges are line segments with length one that do not cross each other. That is, it is a graph that has an embedding which is simultaneously a unit distance graph...
. It has 104 edges and 52 vertices.
In connection with the Happy Ending problem
Happy Ending problem
The Happy Ending problem is the following statement:This was one of the original results that led to the development of Ramsey theory....
, Harborth showed that, for every finite set of ten or more points in general position in the plane, some five of them form a convex pentagon that does not contain any of the other points.
Harborth's Conjecture posits that every planar graph admits a straight-line embedding in the plane where every edge has integer length. This open question is a stronger version of Fáry's theorem
Fáry's theorem
In mathematics, Fáry's theorem states that any simple planar graph can be drawn without crossings so that its edges are straight line segments. That is, the ability to draw graph edges as curves instead of as straight line segments does not allow a larger class of graphs to be drawn...
. It is known to be true for cubic graph
Cubic graph
In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs....
s.
In number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, the Stolarsky–Harborth constant is named for Harborth, along with Kenneth Stolarsky.