Tricolorability

Tricolorability is a concept in mathematics and graph theory that refers to the ability to color the vertices of a graph using three colors in such a way that no two adjacent vertices share the same color. This concept is closely related to the four color theorem, which states that any planar graph can be colored using no more than four colors.

Definition

In graph theory, a branch of mathematics, a graph is said to be tricolorable if its vertices can be colored with three colors in such a way that no two adjacent vertices share the same color. This is equivalent to the graph being bipartite, as a bipartite graph can always be tricolored.

Properties

Tricolorability has several important properties in graph theory. For example, a graph is tricolorable if and only if it is bipartite. This is because a bipartite graph can always be colored with two colors, and adding a third color does not change this property.

Another important property of tricolorable graphs is that they are always planar. This means that they can be drawn on a plane without any edges crossing. This property is closely related to the four color theorem, which states that any planar graph can be colored with no more than four colors.

Applications

Tricolorability has applications in various fields, including computer science, operations research, and scheduling. For example, in computer science, tricolorability can be used to solve problems related to resource allocation and scheduling. In operations research, tricolorability can be used to solve problems related to network design and optimization.

See also

• Four color theorem
• Graph coloring
• Bipartite graph
• Planar graph