Vadim G. Vizing - Research Results

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. 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). 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.

Vizing also made other contributions to graph theory and graph coloring, including the introduction of list coloring, 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, Vizing's conjecture (also unsolved) concerning the domination number of cartesian products of graphs, and the 1974 definition of the modular product of graphs as a way of reducing subgraph isomorphism problems to finding maximum cliques in graphs. 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.