View source for Anton Kotzig

{{short description|Slovak-Canadian mathematician}}
{{More citations needed|date=September 2020}}
{{Infobox scientist
| name              = Anton Kotzig
| image             = <!--(filename only)-->
| image_size        = 
| caption           = 
| birth_date        = {{Birth date|1919|10|22|df=y}}
| birth_place       = [[Kočovce]], [[First Czechoslovak Republic|Czechoslovak Republic]]
| death_date        = {{Death date and age|1991|04|20|1919|10|22|df=y}}
| death_place       = [[Montreal|Montreal, Quebec]], Canada
| nationality       = [[Slovak people|Slovak]]–[[Canadians|Canadian]]
| fields            = [[Mathematics]]
| workplaces        = 
| alma_mater        = [[Comenius University in Bratislava]]
| doctoral_advisor  = 
| doctoral_students =
| known_for         = 
| awards            = 
}}
'''Anton Kotzig''' (22 October 1919 – 20 April 1991) was a [[Slovak people|Slovak]]–[[Canadians|Canadian]] [[mathematician]], expert in [[statistics]], [[combinatorics]] and [[graph theory]].

A number of his mathematical contributions are named after him. These include the [[Ringel–Kotzig conjecture]] on [[graceful labeling]] of [[Tree (graph theory)|trees]] (with [[Gerhard Ringel]]); [[Kotzig's conjecture]] on regularly path connected graphs; [[Kotzig's theorem]] on the [[Degree (graph theory)|degrees of vertices]] in [[convex polyhedron|convex polyhedra]]; as well as the [[Kotzig transformation]].

== Biography ==
Kotzig was born in [[Kočovce]], a village in Western [[Slovakia]]. He studied at the secondary grammar school in [[Nové Mesto nad Váhom]], and began his undergraduate studies at the [[Charles University]] in [[Prague]]. After the closure of [[Czech Republic|Czech]] universities in 1939, he moved to [[Bratislava]] where in 1943, he earned a [[doctoral degree]] (RNDr.) in [[mathematical statistics|Mathematical Statistics]] from the [[Comenius University in Bratislava|Comenius University]].<ref>{{MathGenealogy |id=89823}}</ref> 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]]), where he served as [[Rector (academia)|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]] degree (DrSc.) from the Charles University in 1961 for a thesis on [[Graph theory|Graph Theory]] (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.{{citation needed|date=September 2020}}

In 1969, Kotzig moved to Canada and spent a year at the [[University of Calgary]]. He became a researcher at the Centre de recherches mathematiques (CRM) and the [[Université de Montréal|University of Montreal]] in 1970, where he remained until his death. Because of the political situation, he could not travel back to [[Czechoslovakia]], 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]], 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 graph]]s. 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]]s, special decompositions into [[Hamiltonian path]]s, [[Latin squares]], decompositions of complete graphs, perfect systems of [[difference sets]], additive sequences of [[permutations]], tournaments and  [[Combinatorial game theory|combinatorial games theory]].

[[File:Triakisicosahedron.jpg|thumb|upright|The [[triakis icosahedron]], a polyhedron in which every edge has endpoints with total degree at least 13]]
One of his results, known as [[Kotzig's theorem|Kotzig's Theorem]], is the statement that every [[polyhedral graph]] has an edge whose two endpoints have total [[degree (graph theory)|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]] in the mid-1970s.<ref>{{citation
 | last = Grünbaum | first = Branko
 | contribution = Polytopal graphs
 | mr = 0406868
 | pages = 201–224
 | series = MAA Studies in Mathematics
 | title = Studies in graph theory, Part II
 | volume = 12
 | year = 1975}}; {{citation
 | last = Grünbaum | first = Branko
 | contribution = New views on some old questions of combinatorial geometry
 | mr = 0470861
 | pages = 451–468
 | series = Atti dei Convegni Lincei
 | title = Colloquio Internazionale sulle Teorie Combinatorie (Rome, 1973), Tomo I
 | volume = 17
 | year = 1976}}</ref>

Kotzig published many open problems. One of them is the [[Ringel–Kotzig conjecture]], stating that all trees have a [[graceful labeling]]. In 1963, [[Gerhard Ringel]] proposed that the [[complete graph]] <math>K_{2n+1}</math> could be decomposed into isomorphic copies of any given <math>n</math>-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.<ref>{{citation
 | last = Bloom | first = Gary S.
 | contribution = A chronology of the Ringel–Kotzig conjecture and the continuing quest to call all trees graceful
 | doi = 10.1111/j.1749-6632.1979.tb17766.x
 | mr = 557885
 | pages = 32–51
 | publisher = New York Academy of Sciences | location = New York
 | series = Annals of the New York Academy of Sciences
 | title = Topics in Graph Theory (New York, 1977)
 | volume = 328
 | year = 1979| s2cid = 84953734
 }}</ref> The question remains unsolved.

==Recognition==
In honor of Kotzig's 60th birthday, Alexander Rosa, Gert Sabidussi and Jean Turgeon edited a [[festschrift]], ''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.<ref>{{MR|0806960}}</ref>

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==
{{reflist}}

== External links ==
* [http://hdl.handle.net/10338.dmlcz/129622 Anton Kotzig, 1919–1991], ''Mathematica Slovaca'' 42:3 (1992) 381–383.
* [http://www.mat.savba.sk/MATEMATICI/matematici.php?cislo=117 Prof. Anton Kotzig's biography] (in [[Slovak language|Slovak]]).

{{Authority control}}

{{DEFAULTSORT:Kotzig, Anton}}
[[Category:Slovak mathematicians]]
[[Category:20th-century Canadian mathematicians]]
[[Category:Graph theorists]]
[[Category:Academic staff of the Université de Montréal]]
[[Category:Charles University alumni]]
[[Category:Comenius University alumni]]
[[Category:Czechoslovak emigrants to Canada]]
[[Category:People from Nové Mesto nad Váhom District]]
[[Category:1919 births]]
[[Category:1991 deaths]]
[[Category:Czechoslovak mathematicians]]