The Ringel-Kotzig conjecture on graceful labeling of trees is named after him and Gerhard Ringel.

Kotzig was born in Kočovce, a village in Western Slovakia, in 1919. He studied at the secondary grammar school in Nové Mesto nad Váhom, and began his undergraduate studies at Charles University in Prague. After the closure of Czech universities in 1939, he moved to Bratislava, where after the war he earned a doctoral degree (RNDr.) in mathematical statistics from Comenius University in Bratislava. He remained in Bratislava working at the Central Bureau of Social Insurance for Slovakia, as the head of 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 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 the 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 Applied Mathematics on Faculty of Natural Sciences of Comenius University, where he was also dean for one year. He also earned a habilitation degree (DrSc.) from Charles University in 1961 for a thesis in graph theory (relation and regular relation of finite graphs).

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. One of his arguments is generally known as Kotzig's Theorem. Kotzig established the now well-known Slovak School of Graph Theory. One of his first students was RNDR. Juraj Bosák, DrSc., who was awarded the Czechoslovak State Prize in 1969.

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 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. The result was more than 75 articles, supervision of many masters and doctoral theses, as well as university lecture courses.

His publications covers a wide range of topics in graph theory and combinatorics;; convex polyhedra, quasigroups, special decompositions Hamiltonian paths , Latin squares, decompositions of complete graphs, perfect systems of difference sets, additive sequences of permutations, tournaments and combinatorial games theory.

In honor of his 60th birthday the publication "Theory and Practice of Combinatorics“ was published, to which various experts from around the world contributed. Kotzig published many open problems. One of them is the still unresolved Ringel-Kotzig hypothesis. He died on April 20. 1991 in Montreal, leaving his wife Edita, and a son Ľuboš.

In 1999 a commemorative plaque was erected on his birth house in Kočovce on the 80th anniversary of his birth.