Graph TheoryChapter 4 sec. 1

Koenigsberg bridge problemIt is the Pregel River divided Koenigsberg into four distinct sections. Seven bridges connected the four portions of Koenigsberg.

It was a popular pastime for the citizens of Koenigsberg to start in one section of the city and take a walk visiting all sections of the city, trying to cross each bridge exactly once and to return to the original starting point.

How did it start?In 1735, a Swiss Mathematician Leonhard Euler became the first person to work in graph theory by solving the Koenigsburg bridge problem.

Discovered a simple way to determine when a graph can be traced.

DefinitionTrace-to begin at some vertex and draw the entire graph without lifting your pencil and without going over any edge more than once.

On a piece of paper draw these 2 pictures.Chap 4 sec 1

a) b)

Exercise 1Place your pencil on any dot and trace the figure completely without lifting your pencil and without tracing any part of any line twice.

Which of the two can be done?

Solution

Fig. A can be traced.

Fig. B cannot be traced.

DefinitionsGraph- consists of a finite set of pointsVertices – are points on the graphEdges- are lines that join pairs of verticesConnected- if it is possible to travel from

any vertex to any other vertex of the graph by moving along successive edges.

Bridge- in a connected graph is an edge such that if it were removed the graph is no longer connected.

A A E

C D

B F

Connected graph, the vertex CD is a bridge

A

D E

B C

Nonconnected graph

Odd and Even VertexOdd – The graph is odd if it is an endpoint of an odd number of edges of the graph.

Even- The graph is even if it is an endpoint of an even number of edges of the graph.

Determine which vertices are even and which are odd.

A

B D C

Solution

Vertex A is oddVertex B is oddVertex C is oddVertex D is odd

Determine which vertices are odd and even

A

C D

B

Solution

Vertex A is oddVertex B is oddVertex C is evenVertex D is even

Euler’s TheoremA graph can be traced if it is connected and has zero or two odd vertices.

Which of the graphs can be traced? A B A B

C

C

D E

D

Solution

Fig. 1 Cannot be traced. (all odd)

Fig. 2 Can be traced by Euler’s theorem.

NoteIf a graph has 2 odd vertices, the tracing must begin at one of these and end at the other.

If all vertices are even, then the graph tracing must begin and end at the same vertex. It does not matter at which vertex this occurs.

DefinitionsPath- in a graph is a series of consecutive edges in which no edge is repeated.

Euler path- A path containing all the edges of a graph.

Euler circuit- An Euler path that begins and ends at the same vertex.

Eulerian graph-A graph with all even vertices contains an Euler circuit

Find Euler’s path and Euler’s circuit for the two fig. below.

A B

A C

B E

D E

C D F G

H I

SolutionFig. 1 (star)

Euler’s path - ADBECAEuler ‘s circuit - ADBECA

Fig. 2 Euler’s path – CABCDEHIDFGEuler’s circuit – There is none, because G and C

are both odd vertices, we must begin at one and end at the other.

What is Euler’s circuit used for?How many of you ride the pubic transportation?

Efficient routes.Map Coloring

Eulerizing a Graph1. The graph must have all even vertices.

2. If a graph has an odd vertex, then we will add some edges to make that vertex an even vertex.

3. We want to begin and end at the same vertex.

4. We do not want to travel on the same edge twice.

Find and efficient route.

Thank you