Vadim Georgievich Vizing (Russian: Вади́м Гео́ргиевич Визинг, Ukrainian: Вадим Георгійович Візінг; born 1937) is a Ukrainian (former Soviet) mathematician known for his contributions to graph theory, and especially for Vizing's theorem stating that the edges of any graph with maximum degree Δ can be colored with at most Δ + 1 colors.

## Biography

Vizing was born in Kiev on March 25, 1937.[1][2] His mother was half-German, and because of this the Soviet authorities forced his family to move to Siberia in 1947. After completing his undergraduate studies in mathematics in Tomsk State University in 1959, he began his Ph.D. studies at the Steklov Institute of Mathematics in Moscow, on the subject of function approximation, but he left in 1962 without completing his degree.[1] Instead, he returned to Novosibirsk, working from 1962 to 1968 at the Russian Academy of Sciences there and earning a Ph.D. in 1966.[1] After holding various additional positions, he moved to Odessa in 1974, where he taught mathematics for many years at the Academy for Food Technology.[1]

## Research results

The result now known as Vizing's theorem, published in 1964 when Vizing was working in Novosibirsk, states that the edges of any graph with at most Δ edges per vertex can be colored using at most Δ + 1 colors.[3] It is a continuation of the work of Claude Shannon, who showed that any multigraph can have its edges colored with at most (3/2)Δ colors (a tight bound, as a triangle with Δ/2 edges per side requires this many colors).[4] Although Vizing's theorem is now standard material in many graph theory textbooks, Vizing had trouble publishing the result initially, and his paper on it appears in an obscure journal, Diskret. Analiz.[5]

Vizing also made other contributions to graph theory and graph coloring, including the introduction of list coloring,[6] the formulation of the total coloring conjecture (still unsolved) stating that the edges and vertices of any graph can together be colored with at most Δ + 2 colors,[7] Vizing's conjecture (also unsolved) concerning the domination number of cartesian products of graphs,[8] and the 1974 definition of the modular product of graphs as a way of reducing subgraph isomorphism problems to finding maximum cliques in graphs.[9] He also proved a stronger version of Brook's theorem that applies to list coloring.

From 1976, Vizing stopped working on graph theory and studied problems of scheduling instead, only returning to graph theory again in 1995.[1]

## Notes

1. ^
2. ^ Vizing, V. G. (1964), "On an estimate of the chromatic class of a p-graph", Diskret. Analiz. (In Russian) 3: 25–30, MR 0180505.
3. ^ Shannon, Claude E. (1949), "A theorem on coloring the lines of a network", J. Math. Physics 28: 148–151, MR 0030203. In both Gutin & Toft (2000) and Soifer (2008), Vizing mentions that his work was motivated by Shannon's theorem. For the triangle lower bound example, see e.g. Colorful Mathematics.
4. ^ The full name of this journal was Akademiya Nauk SSSR. Sibirskoe Otdelenie. Institut Matematiki. Diskretny˘ı Analiz. Sbornik Trudov. It was renamed Metody Diskretnogo Analiza in 1980 (the name given for it in Gutin & Toft (2000)) and discontinued in 1991 [1].
5. ^ Vizing, V. G. (1976), "Vertex colorings with given colors", Diskret. Analiz. (In Russian) 29: 3–10.
6. ^ Vizing, V. G. (1968), "Some unsolved problems in graph theory", Uspehi Mat. Naukno. (In Russian) 23 (6): 117–134, MR 0240000.. In Soifer (2008), Vizing states that he formulated the conjecture in 1964, but by the time it was published in 1968 Behzad had independently posed the same conjecture.
7. ^
8. ^ Vizing, V. G. (1974), "Reduction of the problem of isomorphism and isomorphic entrance to the task of finding the nondensity of a graph", Proc. 3rd All-Union Conf. Problems of Theoretical Cybernetics, p. 124.