A Constructive Proof of Vizing's Theorem

We consider finite graphs with no self-loops and no multiple edges. A graph is valid if all edges incident on a vertex have different colors. We prove Vizing’s Theorem. All the edges of a graph of maximum degree less than N can be colored using N colors so that the graph is valid. We call a color incident on a vertex if an edge incident on that vertex has that color; otherwise, the color is free on that vertex. Since there are N available colors and each vertex degree is less than N , there is a free color on each vertex. Our proof consists of showing how to color an arbitrary uncolored edge of a valid graph (which may require changing the colors of already-colored edges to maintain validity). This procedure can be repeated until all edges are colored. Henceforth, X Y is the uncolored edge that is to be colored.