Anton Kotzig
   HOME

TheInfoList



OR:

Anton Kotzig (22 October 1919 – 20 April 1991) was a Slovak
Canadian Canadians () are people identified with the country of Canada. This connection may be residential, legal, historical or cultural. For most Canadians, many (or all) of these connections exist and are collectively the source of their being ''C ...
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, mathematical structure, structure, space, Mathematica ...
, expert in
statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
,
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
and
graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...
. A number of his mathematical contributions are named after him. These include the Ringel–Kotzig conjecture on
graceful labeling In graph theory, a graceful labeling of a Graph (discrete mathematics), graph with edges is a graph labeling, labeling of its Vertex (graph theory), vertices with some subset of the integers from 0 to inclusive, such that no two vertices share ...
of
trees In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, e.g., including only woody plants with secondary growth, only p ...
(with Gerhard Ringel); Kotzig's conjecture on regularly path connected graphs; Kotzig's theorem on the degrees of vertices in
convex polyhedra In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary surfa ...
; as well as the Kotzig transformation.


Biography

Kotzig was born in Kočovce, a village in Western
Slovakia Slovakia, officially the Slovak Republic, is a landlocked country in Central Europe. It is bordered by Poland to the north, Ukraine to the east, Hungary to the south, Austria to the west, and the Czech Republic to the northwest. Slovakia's m ...
. He studied at the secondary grammar school in Nové Mesto nad Váhom, and began his undergraduate studies at the
Charles University Charles University (CUNI; , UK; ; ), or historically as the University of Prague (), is the largest university in the Czech Republic. It is one of the List of oldest universities in continuous operation, oldest universities in the world in conti ...
in
Prague Prague ( ; ) is the capital and List of cities and towns in the Czech Republic, largest city of the Czech Republic and the historical capital of Bohemia. Prague, located on the Vltava River, has a population of about 1.4 million, while its P ...
. After the closure of
Czech Czech may refer to: * Anything from or related to the Czech Republic, a country in Europe ** Czech language ** Czechs, the people of the area ** Czech culture ** Czech cuisine * One of three mythical brothers, Lech, Czech, and Rus *Czech (surnam ...
universities in 1939, he moved to
Bratislava Bratislava (German: ''Pressburg'', Hungarian: ''Pozsony'') is the Capital city, capital and largest city of the Slovakia, Slovak Republic and the fourth largest of all List of cities and towns on the river Danube, cities on the river Danube. ...
where in 1943, he earned a
doctoral degree A doctorate (from Latin ''doctor'', meaning "teacher") or doctoral degree is a postgraduate academic degree awarded by universities and some other educational institutions, derived from the ancient formalism '' licentia docendi'' ("licence to teach ...
(RNDr.) in
Mathematical Statistics Mathematical statistics is the application of probability theory and other mathematical concepts to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques that are commonly used in statistics inc ...
from the Comenius University. He remained in Bratislava working at the Central Bureau of Social Insurance for Slovakia as head of the Department of Mathematical Statistics. Later, he published a book on economy planning. From 1951 to 1959, he lectured at Vysoká škola Ekonomická (today
University of Economics in Bratislava The Bratislava University of Economics and Business () is the oldest university of economics in Slovakia. History The university was established in 1940 as a private university under the name Vysoká obchodná škola v Bratislave (College of Co ...
), where he served as rector from 1952 to 1958. Thus he spent 20 years in close contact with applications of mathematics. In 1959, he left the University of Economics to become head of the newly-created Mathematical Institute of the Slovak Academy of Sciences, where he remained until 1964. From 1965 to 1969, he was head of the department of the Applied Mathematics on Faculty of the Natural Sciences of Comenius University, where he was also dean for one year. He also earned a
habilitation Habilitation is the highest university degree, or the procedure by which it is achieved, in Germany, France, Italy, Poland and some other European and non-English-speaking countries. The candidate fulfills a university's set criteria of excelle ...
degree (DrSc.) from the Charles University in 1961 for a thesis on
Graph Theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...
(relation and regular relation of finite graphs). Kotzig established the now well-known Slovak School of Graph Theory. One of his first students was Juraj Bosák who was awarded the Czechoslovak State Prize in 1969. In 1969, Kotzig moved to Canada and spent a year at the
University of Calgary {{Infobox university , name = University of Calgary , image = University of Calgary coat of arms without motto scroll.svg , image_size = 150px , caption = Coat of arms , former ...
. He became a researcher at the Centre de recherches mathematiques (CRM) and the University of Montreal in 1970, where he remained until his death. Because of the political situation, he could not travel back to
Czechoslovakia Czechoslovakia ( ; Czech language, Czech and , ''Česko-Slovensko'') was a landlocked country in Central Europe, created in 1918, when it declared its independence from Austria-Hungary. In 1938, after the Munich Agreement, the Sudetenland beca ...
, and remained in his adopted country without his books and notes. Although he was separated from his Slovak students, he continued doing mathematics. He died on April 20, 1991 in
Montreal Montreal is the List of towns in Quebec, largest city in the Provinces and territories of Canada, province of Quebec, the List of the largest municipalities in Canada by population, second-largest in Canada, and the List of North American cit ...
, leaving his wife Edita and son Ľuboš.


Contributions

By 1969, the list of his publications already included over 60 articles and 4 books. Many of his results have become classical, including results about graph relations, 1-factors and cubic graphs. As they were published only in Slovak, many of them remained unknown and some of the results were independently rediscovered much later by other mathematicians. In Canada, he wrote more than 75 additional articles. His publications cover a wide range of topics in graph theory and combinatorics: convex polyhedra,
quasigroup In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure that resembles a group in the sense that " division" is always possible. Quasigroups differ from groups mainly in that the associative and identity element pro ...
s, special decompositions into
Hamiltonian path In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vert ...
s,
Latin squares Latin ( or ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken by the Latins in Latium (now known as Lazio), the lower Tiber area around Rome, Italy. Through the expansion o ...
, decompositions of complete graphs, perfect systems of difference sets, additive sequences of
permutations In mathematics, a permutation of a Set (mathematics), set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example ...
, tournaments and combinatorial games theory. One of his results, known as Kotzig's Theorem, is the statement that every
polyhedral graph In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the Vertex (geometry), vertices and Edge (geometry), edges of a convex polyhedron. Alternatively, in purely graph-theoretic terms, the polyh ...
has an edge whose two endpoints have total degree at most 13. An extreme case is the triakis icosahedron, where no edge has smaller total degree. Kotzig published the result in Slovakia in 1955, and it was named and popularized in the West by
Branko Grünbaum Branko Grünbaum (; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentRingel–Kotzig conjecture, stating that all trees have a
graceful labeling In graph theory, a graceful labeling of a Graph (discrete mathematics), graph with edges is a graph labeling, labeling of its Vertex (graph theory), vertices with some subset of the integers from 0 to inclusive, such that no two vertices share ...
. In 1963, Gerhard Ringel proposed that the
complete graph In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices i ...
K_ could be decomposed into isomorphic copies of any given n-vertex tree, and in 1966, Alexander Rosa credited Kotzig with the suggestion that a stronger decomposition always existed, equivalent to the existence of a graceful labeling. The question remains unsolved.


Recognition

In honor of Kotzig's 60th birthday, Alexander Rosa, Gert Sabidussi and Jean Turgeon edited a
festschrift In academia, a ''Festschrift'' (; plural, ''Festschriften'' ) is a book honoring a respected person, especially an academic, and presented during their lifetime. It generally takes the form of an edited volume, containing contributions from the h ...
, ''Theory and Practice of Combinatorics: A collection of articles honoring Anton Kotzig on the occasion of his sixtieth birthday'' (Annals of Discrete Mathematics 12, North-Holland, 1982), with contributions from experts from around the world. In 1999, a commemorative plaque was erected on his birth house in Kočovce on the 80th anniversary of his birth.


See also

* Kotzig transformations * Ringel-Kotzig conjecture


References


External links


Anton Kotzig, 1919–1991
''Mathematica Slovaca'' 42:3 (1992) 381–383.
Prof. Anton Kotzig's biography
(in Slovak). {{DEFAULTSORT:Kotzig, Anton Slovak mathematicians 20th-century Canadian mathematicians Graph theorists Academic staff of the Université de Montréal Charles University alumni Comenius University alumni Czechoslovak emigrants to Canada People from Nové Mesto nad Váhom District 1919 births 1991 deaths Czechoslovak mathematicians