Spring 2015Mathematics in

Management Science

Routing Problems

ExamplesEuler Circuit ProblemsUnicursal DrawingsGraph Theory

Routing Problems

Examples of routing problems• Mail/Garbage collection• Meter reading• Snow plowing• Security patrol

Routing Problems

Existence questionIs an actual route possible?

Optimization questionOf all the possible routes, which one is the optimal route?

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!

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.

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.

Example

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.

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.

Euler Circuit ProblemQuestions:• 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,…

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?