BY: MIKE BASHAM,

Math in Scheduling

The Bridges of Konigsberg

In Konigsberg, Germany there was a river that ran through the city with an island in the middle of the river. Seven bridges were built on the river so people could easily get from one part to another. The people had wondered whether or not it was possible to get around the city in a way that would allow them to cross each of the seven bridges only once.

Euler’s Solution

Euler approaches this problem by labeling the sections of land separated by the water with capital letters. Mathematicians call this process, “labeling the vertices.” Vertices are the plural form of vertex, which is a point where two or more straight lines meet. In the figure shown below, the vertices are labeled by red dots and the edges are represented by the black lines. Euler solves this problem by drawing a figure similar to the one below without retracing any line, and making sure to keep his writing utensil on the paper.

D E F I N I T I O N : A N E T W O R K I S A F I G U R E M A D E U P O F P O I N T S ( V E RT I C E S ) C O N N E C T E D BY N O N - I N T E R S E C T I N G C U R V E S

( A R C S ) . 

D E F I N I T I O N : A V E R T E X I S C A L L E D O D D I F I T H A S A N O D D N U M B E R O F A R C S L E A D I N G T O I T, O T H E R W I S E I T I S C A L L E D

E V E N . 

D E F I N I T I O N : A N E U L E R PAT H I S A C O N T I N U O U S PAT H T H AT PA S S E S T H R O U G H E V E R Y A R C O N C E A N D O N LY O N C E .

Graph Theory Euler uses this way of thinking to lay the foundation for

the Graph Theory. Below we have listed the three definitions that prove that theory.

Graph Theory Continued…

Theorem: If a network has more than two odd vertices, it does not have an Euler path.

Theorem: If a network has two or zero odd vertices, it has at least one Euler path. In particular, if a network has exactly two odd vertices, then its Euler paths can only start on one of the odd vertices, and end on the other.

Euler also provides these two rules.

WHAT IS A CIRCUIT? WHAT IS THE DIFFERENCE BETWEEN EULER AND

HAMILTON’S CIRCUITS? IS THE KONIGSBERG PROBLEM UNSOLVABLE?

Circuits

What is a Circuit?

Circuit- a circuit is a closed, simple path.Paths- a path in a graph is a sequence of

vertices such that from each of its vertices there is an edge to the next vertex in the sequence.

Edges- An edge is a line segment that joins two vertices.

Vertices- A vertex (plural: vertices) is a point where two or more straight lines meet. Basically it’s a corner.

Euler’s Circuit

A Eulerian circuit or Eulerian cycle is a path on a graph that starts and ends of the same vertex.

A Eulerian Circuit uses each edge exactly once.

Leonhard Euler first explored this idea in 1736 with his Seven Bridges of Konigsberg Problem.

It is not a Eulerian Circuit unless all of it’s vertices have an even degree.

Fleury and Hierholzer’s Algorithm

Leonhard Euler never developed an algorithm for his mathematical theory however there were two mathematicians that did.

Hierholzer established his algorithm first in 1873. It is said to be the more efficient of the two.

Fleury developed his algorithm in 1883, and although it recieves criticism, it is still a method that works.

Fleury’s Algorithm

Consider a graph known to have all edges in the same component and at most two vertices of odd degree. We start with a vertex of odd degree—if the graph has none, then start with any vertex. At each step we move across an edge whose deletion would not disconnect the graph, unless we have no choice, then we delete that edge. At the end of the algorithm there are no edges left, and the sequence of edges we moved across forms an Eulerian cycle if the graph has no vertices of odd degree; or an Eulerian trail if there are exactly two vertices of odd degree.

Hierholzer’s Algorithm

• Choose any starting vertex v, and follow a trail of edges from that vertex until returning to v. It is not possible to get stuck at any vertex other than v, because the even degree of all vertices ensures that, when the trail enters another vertex w there must be an unused edge leaving w. The tour formed in this way is a closed tour, but may not cover all the vertices and edges of the initial graph.

• As long as there exists a vertex v that belongs to the current tour but that has adjacent edges not part of the tour, start another trail from v, following unused edges until returning to v, and join the tour formed in this way to the previous tour.

By using a data structure such as a doubly linked list to maintain the set of unused edges incident to each vertex, to maintain the list of vertices on the current tour that have unused edges, and to maintain the tour itself, the individual operations of the algorithm (finding unused edges exiting each vertex, finding a new starting vertex for a tour, and connecting two tours that share a vertex) may be performed in constant time each, so the overall algorithm takes linear time

Hamilton Circuit

Hamilton circuits are named after the renowned Irish mathematician (and astonomer) Sir William Rowan Hamilton who lived from 1805 to 1865. Hamilton was a child prodigy. He could read English, Hebrew, Greek, and Latin by time he was four years old. In addition, he wrote poetry and maintained close friendships with Wordsworth and Coleridge. Hamilton became Professor of Astronomy at Trinity College in Dublin, Ireland when he was twenty-three years old.

A circuit that starts at a vertex of a graph, passes through every vertex exactly once, and returns to the starting vertex is a HAMILTON CIRCUIT.

If a graph contains a Hamiltonian circuit is it considered a Hamiltonian graph.

Dodecahedron (Hamiltonian Circuit)

Euler Circuits Hamiltonian Circuits

All edges of graph are traversed exactly once.

All vertices HAVE to be of even degree.

In these circuits you only have to pass through every vertex once and end on the same vertex on which you started.

The vertices do not need to be of even degree.

The “Optimal Hamiltonian Path,” is that in which the least slides are used.

The Difference Between Euler Circuits and Hamiltonian Circuits

Quick Examples to Practice

Can you take a walk through the town, visiting each part of the town and crossing each bridge only once?

To "visit each part of the town" the person taking the walk should visit the points A, B, C and D. And they should cross each bridge p, q, r, s, t, u and v just once.

It can be further simplified in a graph like this. See if you can solve it.

Why is the Konigsberg Problem Unsolvable?

Vertices A, B and D have degree 3 and vertex C has degree 5, so this graph has four vertices of odd degree. Therefore, it does not have an Euler Path. However, it does have a Hamiltonian Path.