1. Yu-Jung Liang Department of Applied Mathematics National Dong Hwa University Advisor: Professor David Kuo

2. Outline Introduction Previous Results Main Results

3. Definition (The Cartesian product)

5. Definition (Rainbow connection number)

6. 1 2 34 5 234 5 12 3 45 1 1 2 31 2 231 2 12 3 12 1

7. Previous Results back1

8. back

9. Main Results Particular labeling 0 0 1 1 1 2 2 3 2 3 3 0 4 0 0 1 1 1 2 2 3 2 3 3 0 4 1 0

10. Rainbow connection numbers of the Cartesian product of paths and cycles

11. 1 2 31 3 231 3 12 3 12 1 1 2 2 12 3 1 Thm 1

12. Rainbow connection numbers of the Cartesian product of two trees

13. 1 2 3 1 2 3 1 3 Thm3

15. 1 1 2 2 33 44 5 6 7 6

16. 4 4 5 5 66 11 2 3 7 3

17. 1 1 2 2 3 3 7 44 5 6 8 7 7 8 4 4 1 1 2 3 3 5 5 6 6 7 8 8 8 8 8 8 8 8

18. 1 2 3 4 5 6 6 78 1 2 3 4 5 6 3 2 1 3

19. 4 5 6 1 2 3 3 78 4 5 6 1 2 3 6 5 4 6

20. 1 2 3 4 5 6 6 78 9 1 2 3 7 8 7 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 7 8 4 4 1 1 5 66 4 5 6 6 5 2 2 3 3 3

21. 1 2 3 1 2 3 1 23 1

22. 1 1 12 2 3 2 1 1 12 3 1 3 3 3 12 2 3 2 1 2 3 1 2 3 1 23 1