Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Euler Path and Circuit - Lecture Notes | MGF 1107, Study notes of Mathematics for Liberal Arts

Material Type: Notes; Class: Math for Liberal Arts; Subject: MGF: Math - General & Finite; University: Valencia Community College; Term: Unknown 1989;

Typology: Study notes

2009/2010

Uploaded on 02/25/2010

koofers-user-qyo
koofers-user-qyo 🇺🇸

10 documents

1 / 2

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
1
6.1 Euler Path, Circuit
INSTRUCTOR NOTES
LESS THAN 1 hour 15 min. (continue with section 6.2)
CLASS DO: Have them try Konigsberg problem on p. 364.
Then show graph representation of Konigsberg.
KONIGSBERG GRAPH:
Draw graphs below on board. Tell students to review at home all bold-faced
vocabulary words on pages 364-5.
EULER PATH:
Is a path between two vertices which passes each edge exactly once
.
EULER CIRCUIT: Is an Euler Path that starts and ends at the same vertex.
EVEN VERTEX: A vertex that has an even number of edges coming out of it.
ODD VERTEX: A vertex that has an odd number of edges coming out of it.
Label vertices on graphs above as either even or odd.
NOTE: It is a fact that odd vertices come in pairs.
Note for instructor: p. 379 # 32 asks for a proof of this fact: reference Solutions Manual.
TELL CLASS If an Euler Path exists: Look at a vertex that is not an end or a
beginning. That vertex must be even because if a path goes to the vertex then the
path must also leave the vertex.
This idea helps us to understand the Euler theorem.
pf2

Partial preview of the text

Download Euler Path and Circuit - Lecture Notes | MGF 1107 and more Study notes Mathematics for Liberal Arts in PDF only on Docsity!

1

6.1 Euler Path, Circuit

INSTRUCTOR NOTES

LESS THAN 1 hour 15 min. (continue with section 6.2)

CLASS DO : Have them try Konigsberg problem on p. 364. Then show graph representation of Konigsberg.

KONIGSBERG GRAPH:

Draw graphs below on board. Tell students to review at home all bold-faced vocabulary words on pages 364-5.

EULER PATH : Is a path between two vertices which passes each edge exactly once. EULER CIRCUIT : Is an Euler Path that starts and ends at the same vertex. EVEN VERTEX : A vertex that has an even number of edges coming out of it. ODD VERTEX : A vertex that has an odd number of edges coming out of it.

Label vertices on graphs above as either even or odd. NOTE: It is a fact that odd vertices come in pairs. Note for instructor: p. 379 # 32 asks for a proof of this fact: reference Solutions Manual. TELL CLASS If an Euler Path exists: Look at a vertex that is not an end or a beginning. That vertex must be even because if a path goes to the vertex then the path must also leave the vertex.

This idea helps us to understand the Euler theorem.

2

p. 369 Euler Theorem :

  • A connected graph has an Euler Circuit if all vertices are even.
  • A connected graph has an Euler Path there are exactly two odd vertices OR if all of the vertices are even.
  • A graph has neither an Euler Path nor an Euler Circuit if there are more than two odd vertices.

Revisit graphs on first page. Give Euler paths/circuits if they exist (use directional arrows with circled numbers).

CLASS DO : p. 376 # 8, 10

YOU DO : p. 378 # 22 ANSWER for part a) A B C

D E F

G

YOU DO : p. 373 apply Eulerization definition to

YOU DO : p. 378 # 24 NOTE that edges must be added along existing edges.