1. Examples Euler Circuit Problems Unicursal Drawings Graph Theory
Routing Problems
Examples
Euler Circuit Problems
Unicursal Drawings
Graph Theory
Mathematics in Management Science
Spring 2015

2. Routing Problems Examples of routing problems Mail/Garbage collection
Meter reading
Snow plowing
Security patrol

3. Routing Problems Existence question Is an actual route possible?
Optimization question
Of all the possible routes, which one is the optimal route?

4. Euler Circuit Problem These are special Routing Problems.
The common thread is so-called exhaustion requirement;
the route must ‘go everywhere’.
Typically want to begin & end at same location.
We’ll see that ECPs are easy to solve!

5. Unicursal Drawings
Want to trace each drawing w/o lifting pencil or retracing any of the lines.
If we begin & end in the same place, call it a closed unicursal tracing; otherwise an open unicursal tracing.

6. Graphs A graph is a finite set of dots and connecting links.
The dots are called vertices.
The links are called edges.
The valence of a vertex is the number edges that meet there.
Vertices can be even or odd depending on their valence.

7. Example

8. Paths and Circuits
A path is a finite sequence of adjacent edges on a graph that joins two vertices.
A circuit is a path that begins and ends at the same vertex.
A graph is connected if every vertex can be joined to every other vertex by a path.

9. Euler Paths and Circuits
An Euler path is a path that traverses each edge exactly once.
An Euler circuit is a circuit that traverses each edge exactly once.

11. Euler Circuit Problem Questions: Do all graphs have Euler circuits?
Answer: No.
Can we tell which graphs have an Euler circuit and which don’t?
Answer: Yes!
First, the graph must be all one piece--- it must be connected. Next,…

12. Euler’s Circuit Theorem
A graph has an Euler circuit
if and only if
the graph is connected,
and,
all its vertices have even valence.
Any odd vertex means NO Euler circuit.
Once we know they exist:
How do we find Euler circuits?