1. Introduction to Euler graph Nguyen Ngoc Trung Hochiminh University of Pedagogy trung_2n@yahoo.com

2. Introduction to Euler graph Introduction Process Evaluation Conclusion

3. Introduction Euler graph plays an very important role in many applications of graph. You will know concept of Euler graph after this lesson. Properties of Euler graph are also explored within this lesson. Some more research on Euler have to be done by students. Back to main page

4. Process Activity 1. Games Consider two figures, can you draw them by a continuous line, starting and finishing at a same point? Back to main page Figure 1. Figure 2.

5. Answer!!!! Only Figure 2 can be drawn.

6. Process (cont) Activity 2. Study concept of Euler graph Figure 2 is an example of Euler graph. An Euler graph is a graph that has a single circuit pass through all its edges. Back to main page Figure 1. Figure 2.

7. Process (cont) Activity 3. Point out properties of Euler graph What is the difference between Fig 1. and Fig. 2 that Fig 2 can not be drawn? Hint: Attention to degree of vertices!!!! Back to main page Figure 1. Figure 2.

8. Answer!!! All vertices of graph in Fig. 2 have even degree while graph in Fig. 1 has two vertices that have odd degree. Back to main page Figure 1. Figure 2.

9. Process (cont) Activity 3. (cont) Conjecture: An Euler graph has all its vertices which have even degree??? And vice versa? That is true. Go to website: http://www.math.uci.edu/~mathcirc/math194/lectures/graphs/node2.html and discover the theorem and the proof. Documents: www.math.unm.edu/~loring/links/discrete_f05/euler.pdf www.math.unm.edu/~loring/links/discrete_f05/euler.pdf

10. Process (cont) Activity 4. How to find Euler circuit? Back to Fig. 2. How can it be drawn? Fail!!!

11. Process (cont) Activity 4. How to find Euler circuit? Visit following website to watch the demonstration of finding Euler circuit http://www.people.vcu.edu/~gasmerom/MAT131/euler.html Task (homework): 3 groups (4 students each group): Group 1 - 3 Research algorithm for finding Euler circuit and present next time in class. Build a program in C/C++ for finding Euler circuit based on researched algorithm. Expected products: a report and a program

12. Process (cont) Activity 4. (cont) Resources: http://portal.acm.org/citation.cfm?id=808688&coll=portal&dl= ACM http://portal.acm.org/citation.cfm?id=808688&coll=portal&dl= ACM www.engin.umd.umich.edu/CIS/course.des/cis350/notes/page3 5.html www.engin.umd.umich.edu/CIS/course.des/cis350/notes/page3 5.html www.math.uiuc.edu/~tkilbour/math118-f05/final-review.pdf www.pballew.net/graphs.html

13. Process (cont) Activity 5. Applications of Euler graph? Task (homework): 3 groups (4 students each group): Group 4 - 6 Research and present application of Euler graph. Expected products: A report (5-10 pages) about applications of Euler graph.

14. Process (cont) Activity 5. (cont) Resources: www.geocities.com/harry_robinson_testing/graph_th eory.htm www.geocities.com/harry_robinson_testing/graph_th eory.htm www.cs.uvic.ca/~ruskey/Publications/VennArea/Eule r2004.pdf www.cs.uvic.ca/~ruskey/Publications/VennArea/Eule r2004.pdf http://portal.acm.org/ft_gateway.cfm?id=1221538&t ype=pdf&coll=&dl=ACM&CFID=15151515&CFTOKE N=6184618 http://portal.acm.org/ft_gateway.cfm?id=1221538&t ype=pdf&coll=&dl=ACM&CFID=15151515&CFTOKE N=6184618

15. Evaluation Back to main page 102030Score Follow Direction I didn’t follow at all I read the direction but skipped directions or did not follow them completely. I followed the directions completely and did everything I was asked to do. Understanding concept of Euler graph I didn’t understand at all I can understand a part of concept I understand the concept Understanding property of Euler graph I didn’t understand at all I got sense about the property but the proof is too difficult for me to understand I understand the property and the proof. Group works I cannot do anything I can understand the theory but cannot program I got expected products

16. Conclusion We call “Euler graph” because it is first developed by Euler – a Swiss Mathematician. At this point, we can understand the concept, property and application of Euler graph. Future works: Hamilton graph Finding shortest path problem … Back to main page