Chapter 5.4, Problem 31ES

Scheduling Edge colorings, as explained in Exercise 30, can be used to solve scheduling problems. For instance, suppose five players are competing in a tennis tournament. Each player needs to play every other player in a match (but not more than once). Each player will participate in no more than one match per day, and two matches can occur at the same time when possible. How many days will be required for the tournament? Represent the tournament as a graph, in which each vertex corresponds to a player and an edge joins two vertices if the corresponding players will compete against each other in a match. Next, color the edges, where each different color corresponds to a different day of the tournament. Because one player will not be in more than one match per day, no two edges of the same color can meet at the same vertex. If we can find an edge coloring of the graph that uses the fewest number of colors possible, it will correspond to the fewest number of days required for the tournament- Sketch a graph that represents the tournament, find an edge coloring using the fewest number of colors possible, and use your graph to design a schedule of matches for the tournament that minimizes the number of days required.