vertexcolored

Vertex coloring is a concept in graph theory where each vertex of a graph is assigned a color such that no two adjacent vertices share the same color. The goal of vertex coloring is often to use the minimum number of colors possible, which is known as the chromatic number of the graph. A valid vertex coloring is essential for many applications, such as scheduling problems, register allocation in compilers, and map coloring.

The process of finding a vertex coloring can be done using various algorithms. Greedy algorithms are common, where vertices are processed in some order and assigned the smallest available color that is not used by any of its already colored neighbors. However, greedy algorithms do not always produce an optimal coloring (i.e., one that uses the chromatic number of colors). Determining the chromatic number of a general graph is an NP-hard problem, meaning there is no known efficient algorithm to solve it for all cases.

Different types of graphs have specific properties regarding vertex coloring. For instance, bipartite graphs can always be vertex colored with just two colors. Complete graphs require a number of colors equal to the number of vertices. Trees are also bipartite and can be colored with two colors. The study of vertex coloring is a fundamental area within graph theory with broad practical implications.