1 / 23

R ainbow connection numbers of Cartesian product of graphs

R ainbow connection numbers of Cartesian product of graphs. Yu-Jung Liang Department of Applied Mathematics National Dong Hwa University Advisor: Professor David Kuo. Outline. Introduction Previous Results Main Results. Definition (The Cartesian product).

yuki
Télécharger la présentation

R ainbow connection numbers of Cartesian product of graphs

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Rainbow connection numbers of Cartesian product of graphs 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) • Give two graphs and , the Cartesian product of and , denoted by , is defined as follows: .Two distinct vertices and of are adjacent if and only if either and or and .

  4. Example:

  5. Definition (Rainbow connection number) • A path is rainbow if no two edges of it are colored the same. • An edge-coloring graph is rainbow connected if any two vertices are connected by a rainbow path. • We define the rainbow connection number of a connected graph , denoted by rc, as the smallest number of colors that are needed in order to make rainbow connected. • A graph is strong rainbow connected if there exists a rainbow geodesic for any two vertices and in .

  6. 2 5 • Example: 2 1 3 1 2 1 3 4 3 1 2 1 2 3 1 2 4 5 1 2 3 1 1 2 3 4 2 5 rc

  7. Previous Results • Theorem 1 (Xueliang Li, Yuefang Sun) For any connected graph . • Theorem 2 (XueliangLi, YuefangSun) Let , where each is connected. Then we have . Moreover, if for each , then the equality holds. back1

  8. Theorem 3 (M. Basavaraju, L.S. Chandran, D. Rajendraprasad, and A. Ramaswamy) If and are non-trivial connected graphs, then . back

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

  10. Rainbow connection numbers of the Cartesian product of paths and cycles • If , then for all . • If , then for all . • If n is even, then , for all .

  11. 3 • Example: 2 2 1 3 1 1 3 3 1 2 1 2 1 2 1 2 3 1 1 2 3 Thm1

  12. Rainbow connection numbers of the Cartesian product of two trees If and are trees, we have follows conclusion about .

  13. If , , then . 1 1 3 2 3 1 3 2 Thm3

  14. If , has three edge-disjoint path with length , then .

  15. 7 • Otherwise, we have . Example 1 1 4 4 5 2 2 3 3 6 6

  16. 7 4 4 1 1 2 5 5 3 3 6 6

  17. 7 4 4 1 1 • Example 7 7 5 2 5 8 8 8 7 8 6 6 3 3 1 1 4 4 8 8 8 5 2 2 8 8 8 3 3 6

  18. 7 8 1 1 • Example 1 4 4 2 5 5 2 2 3 3 3 3 6 6 6

  19. 7 8 4 4 4 1 1 5 2 2 5 5 6 6 6 6 3 3 3

  20. 8 7 4 4 8 • Example 5 4 1 1 7 8 7 1 9 9 7 5 5 9 6 6 1 4 9 9 2 9 2 2 6 6 9 5 9 9 3 2 3 3 9 9 3 9 9 9 9 9 3 6 6

  21. 1 • Example 2 1 3 3 2 1 2 3 1

  22. 2 1 • In fact, if we consider subgraph of , we have three nature graph. Hence easy to check have rainbow path for any vertices in . 3 2 1 2 1 1 2 2 3 1 3 3 2 1 3 2 1 2 3 2 3 1 3 1 1 2 1 3 1

More Related