One problem I have looked at is "what is the ultimate or best schedule scheme for a bus and a bus system?"

What are the functions that a schedule scheme should optimize?

- Reduce the waiting time between transfers between one bus and another bus.
- Provide "meets" during off hours so night time travelers are not marooned at a bus stop when a slightly early bus misses a transfer time with a slightly later bus.
- Organize the rolling stock to carry the most people during rush hours.
- Arrange for the driver to start and end his driving work at the bus terminal.

- Think of each bus route of a bus system as a loop.
- Think of all the bus routes of a bus system as many loops with many loops crossing. Like a pile of rubber bands.
- The place where s bus route crosses another route is called a node.
- The length of bus route between nodes is called an edge.
- As long as there are an even number of bus routes entering and exiting each node, there exists an Euler path.
- An Euler path can be completely driven by a bus driving over each stretch of road (or edge) .
- The Euler path is in honor of the famous problem solved by Leonard Euler called the "Seven Bridges of Konigsberg." A Wikipedia entry for Eulerian Path

- Suppose we look for a timing structure where there is the smallest wait time when transfering from one bus to a different bus.

- If two buses on different routes arrive at the same time and wait for passengers to transfer from the other bus this is the shortest wait for all travelers.
- If the travel time between system nodes is made the same then the most nodes will have the least transfer time in all directions.