Some graph coloring links:

• a good description of applications
• a vertex coloring problem definition.
• an edge coloring problem defintion
• a briefer list of applications.

Vertex coloring applications:

• Many scheduling problems involve allowing for a number of pairwise restrictions on which jobs can be done simultaneously. For instance, in attempting to schedule classes at a university, two courses taught by the same faculty member cannot be scheduled for the same time slot. Similarly, two courses that are required by the same group of students also should not conflict. The problem of determining the minimum number of time slots needed subject to these restrictions is a graph coloring problem. The time slots are colors for the vertices, the vertices are courses, and the edges between courses are restrictions that force different time slots.
• One very active application for vertex coloring is register allocation. The register allocation problem is to assign variables to a limited number of hardware registers during program execution. (This is something every compiler does!) We want toput as many variables in registers as possible for quick execution of the program. There are typically far more variables than registers so it is necessary to assign multiple variables to registers. Variables conflict with each other if one is used both before and after the other within a short period of time (for instance, within a subroutine). The goal is to assign variables that do not conflict so as to minimize the use of non-register memory. What would the vertices and edges represent?

Edge coloring applications:

• Assume we need to schedule a given set of two-person interviews, where each interview takes one hour. All meetings could be scheduled to occur at distinct times to avoid conflicts, but it is less wasteful to schedule nonconflicting events simultaneously. We can construct a graph whose vertices are the people and whose edges represent the pairs of people who want to meet. An edge coloring of this graph defines the schedule. The color classes represent the different time periods in the schedule, with all meetings of the same color happening simultaneously.
• The National Football League solves such an edge coloring problem each season to make up its schedule. Each team's opponents are determined by the records of the previous season. Assigning the opponents to weeks of the season is the edge-coloring problem.

Edge coloring can be reduced to vertex coloring (in linear time) by constructing the line graph of the input graph G. This is the graph constructed by replacing each edge with a vertex, and connected vertices in the new graph according to the edges that share a vertex in the original graph.