Supplement Part II Graph Coloring Applications

To create a graph connect vertices that CANNOT be together. Color the graph using the method given in the first part of the supplement. State which items CAN be grouped together

Day Care The following children are not allowed to play together in the daycare (because they BITE each other!). Use graph coloring to figure out how many rooms are needed to accommodate all the kids. ChildIs not allowed to play with CorrinEric, Ann BethAnn, William EricWilliam SamCorrin, Beth

Day Care (cont’d) Two vertices are adjacent if the children they represent cannot play together. CorrinSam Eric Ann Beth William

Day Care (cont’d) Next we need to color the graph, so we need to: CorrinSam Eric Ann Beth William purple green red Write down the degree of each vertex Make a list of colors

Day Care (cont’d) Next we color the graph using the familiar technique. CorrinSam Eric Ann Beth William purple green red

Day Care (cont’d) Next we color the graph using the familiar technique. CorrinSam Eric Ann Beth William purple green red

Day Care (cont’d) Next we color the graph using the familiar technique. CorrinSam Eric Ann Beth William purple green red

Day Care (cont’d) 3 rooms are needed. The children that CAN be in the same room are: 1 st room: Corrin, Beth 2 nd room: Eric, Anne and Sam 3 rd room: William 2

Groceries A Graph Theory enthusiast needs to buy 8 items from the grocery store and decides to figure out how many bags she will need according to graph coloring principles. Certain foods cannot be bagged together for health and cleanliness reasons. ItemCannot be bagged with raw meatproduce, bread, bleach, eggs, milk, magazine producemilk, raw meat, bread, bleach bleachbread, produce, eggs, milk, raw meat milkbleach, raw meat, magazine, produce, bread canned corn

Groceries (cont’d) Veritces – Items Edges – connect items that cannot be bagged together Produce Meat Bread Magazine Milk Eggs Bleach Canned Corn

Groceries (cont’d) Put degrees on vertices Make a list of colors Since we don’t know how many colors will be needed we can add to the list of colors as we color the graph. Produce Meat Bread Magazine Milk Eggs Bleach Canned Corn purple green red blue yellow

Groceries (cont’d) Color the graph using the familiar technique. Produce Meat Bread Magazine Milk Eggs Bleach Canned Corn purple green red blue yellow

Groceries (cont’d) Color the graph using the familiar technique. Produce Meat Bread Magazine Milk Eggs Bleach Canned Corn purple green red blue yellow

Groceries (cont’d) Color the graph using the familiar technique. Produce Meat Bread Magazine Milk Eggs Bleach Canned Corn purple green red blue yellow

Groceries (cont’d) 5 bags are needed 1 st bag Meat and Canned Corn 2 nd bag Bleach and Magazine 3 rd bag Milk and Eggs 4 th bag Bread 5 th bag Produce